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3

The Body Mass Index is used as an indicator of whether or not people are in a healthy weight range for their height. If B represents the Body Mass Index, m is the person’s mass in kilograms and h is the person’s height in metres,

then .

A person is considered to be in a healthy weight range if 21 ≤ B ≤ 25.

Can you comment on Richard’s weight given that he is 1.75 m tall and has a mass of 86 kg?

Completing this chapter will refresh your skills in using pronumerals to represent unknown numbers and quantities. You will learn how to simplify algebraic expressions and substitute known values into formulas.

Bmh2-----=

Introductoryalgebra

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80

M a t h s Q u e s t 9 f o r V i c t o r i a

Using pronumerals

The language of algebra

Like English, French or a computer language, algebra is a type of language.

When reading a language we learn to recognise the various written parts. Considerthe following algebraic sentence.

4

xy

+

5

x

3

y

=

7

xy

+

y

2 This can be broken down as follows:

Some important words that you need to be familiar with include: equation, expression, term, coefficient and pronumeral.

When asked to find the coefficient of an algebraic term, in general it is the number inthe term. If no number appears, then 1 is assumed to be the coefficient.

1

×

x

=

1

x

=

x

In the case of a fractional term, we consider the numerical fraction as being multi-plied by the pronumeral and, so, the fraction is the coefficient.

Algebra object Name How do we recognise it?

4

xy

CoefficientPronumeralPronumeral

The number part of a termA letter part of the termA letter part of the term

4

xy

+

5

x

3

y

7

xyy

2

TermTermTermTermTermConstant term

A ‘group’ of letters and numbers at the beginning of an expression or separated by either a

+

sign or a

sign: 7

xy

means 7

×

x

×

y.

A term with no pronumeral

4

xy

+

5

x

3

y

7

xy

+

y

2ExpressionExpression

Algebraic expression or term on each side of the equals sign.

4

xy

+

5

x

3

y

=

7

xy

+

y

2 Equation It contains an equals sign.

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a

81

Note

: A term such as

+

3 does not have a pronumeral part. We call this a

constant

term.A constant term has no pronumeral and its value remains unchanged (constant) regard-less of the value of the pronumeral.

Coefficients occur only in terms where there are pronumerals and they are usuallyplaced at the front of the term.

Find the coefficient of each of these algebraic terms.

a 7xy b m c −2p d

THINK WRITE

a Find the number in the term. The parts of the term are 7, x and y.

a The coefficient is 7.

b Find the number in the term. This term means ‘one m’. If no coefficient is written, a 1 is assumed.

b The coefficient is 1.

c Find the number in the term. The parts of the term are −2 and p.

c The coefficient is −2.

d is the same as a. d The coefficient is .

a5---

a5--- 1

5---

15---

1WORKEDExample

Answer the following for each expression below.i State the number of terms.ii State the coefficient of the second term.iii State the constant term (if there is one).iv State the term with the smallest coefficient.a 4x − 5xy + y2 − 3 b 6x − 3xy + z + 2 + x2z

THINK WRITEa Count the number of terms.

The terms are 4x, −5xy, y2 and −3. a There are 4 terms.

Identify the second term (−5xy). The number part is the coefficient.

The coefficient of the second term is −5.

Identify the constant term. It is the term with no pronumeral.

The constant term is −3.

Identify the smallest coefficient and write the whole term to which it belongs.

The term with the smallest coefficient is −5xy.

b Count the number of terms.The terms are 6x, −3xy, z, 2 and x2z.

b There are 5 terms.

Identify the second term (−3xy). The number part is the coefficient.

The coefficient of the second term is −3.

Identify the constant term. The constant term is +2.Identify the smallest coefficient and write the whole term to which it belongs.

The term with the smallest coefficient is−3xy.

1

2

3

4

1

2

3

4

2WORKEDExample

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82 M a t h s Q u e s t 9 f o r V i c t o r i a

What is a pronumeral?A pronumeral is a letter or symbol that is used in place of a number. When we considerterms such as 7x we know from previous years that this means 7 × x. The pronumeral xhas a value which is not currently known or specified.

Pronumerals are used to write general expressions or formulas that will allow us tomake a substitution for the pronumeral when the value becomes known.

When writing a general expression we choose a pronumeral that can easily be iden-tified as belonging to the unknown quantity that it represents.

Write algebraic expressions for each of the following.a A number 6 more than Ben’s ageb The product of a and wc One more than the age difference between Albert and his son Walterd Five times an unknown quantity is added to six times another unknown quantity.

THINK WRITE

a Since Ben’s age is unknown, use a pronumeral.

a Let a = Ben’s age.

Six more means ‘add 6’. The number is a + 6.b ‘Product’ means multiply. b awc Choose pronumerals to represent

Albert’s age and Walter’s age.c Let a = Albert’s age.

Let w = Walter’s age.The age difference between Albert and Walter is a − w. Add 1 more to this difference.

a − w + 1

d Choose pronumerals for the 2 unknown quantities.

d Let x = the first unknown quantity.Let y = the second unknown quantity.

The sentence can be broken into 3 instructions:5 times an unknown quantity (5x). . . is added to . . . (+)6 times another unknown quantity (6y).

5x + 6y

1

2

1

2

1

2

3WORKEDExample

remember1. A pronumeral is a letter or symbol that is used in place of a number.2. A term is a combination of a number and pronumerals.3. The number part of a term is called the coefficient.4. A term that does not contain a pronumeral part is called a constant.5. An expression is a group of terms separated by + or − signs.6. An equation is a mathematical sentence that puts two expressions equal to each

other.7. When writing expressions, think about which operations are being used, and

the order in which they occur.8. If pronumerals are not given in a question, choose an appropriate letter to use.

remember

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a 83

Using pronumerals

1 Find the coefficient of each of the following terms.a 3x b 7a c −2m d −8q e w

f −n g h i − j −

2 In each of the following expressions state the coefficient of x.a 6x − 3y b 5 + 7x c 5x2 + 3x − 2 d −7x2 − 2x + 4 e 3x − 2x2 f −9x2 − 2x

g 5x2 + 3 − 7x h −11x + 5 − 2x2 i 1 −

j + + k x3 + x + 4 l 2x2 − 5

3 Answer the following for each expression below.i State the number of terms.ii State the coefficient of the first term.iii State the constant term (if there is one).iv State the term with the smallest coefficient.

a 5x2 + 7x + 8 b −9m2 + 8m − 6 c 5x2y − 7x2 + 8xy + 5 d 9ab2 − 8a − 9b2 + 4e 11p2q2 − 4 + 5p − 7q − p2 f −9p + 5 − 7q2 + 5p2q + qg 4a − 2 + 9a2b2 − 3ac h 5s + s2t + 9 + 12t − 3ui −m + 8 + 5n2m + m2 + 2n j 7c2d + 5d2 + 14 − 3cd2 − 2e

4 Write algebraic expressions for each of the following:a a number 2 more than p b a number 7 less than qc 2 is added to 3 times p d 7 is subtracted from 9 times qe 4 times p is subtracted from 10 f 5 minus 2 times pg the sum of p and q h the difference between p and qi 3 times p is added to q j 2 times q is subtracted from pk the product of p and q l 4 times the product of p and qm the sum of 2 times p and 3 times q n 3 times p is subtracted from 2 times qo p is divided by two times q p 3q is divided by p.

5a There are 27 students in the classroom and x are called out to see the principal. The

number remaining in the room is:

b If y people enter a shop where there are 11 customers and 2 sales assistants, thenumber of people in the shop is:

c If a packet of Smarties contains p Smarties, and they are to be divided up between4 people, the number of Smarties each person receives is:

d If a T-shirt costs n dollars, ten t-shirts would cost:

A 27x B 27 − x C x − 27 D 27 + x E

A y + 11 B y − 13 C 13y D 13 + y E

A B 4p C 4 + p D 4 − p E p − 4

A n + 10 B 10n C D 10n + 10 E 10 − n

3AWWORKEDORKEDEExample

1

Mathcad

Displaying thecoefficients DIY

x3--- y

2--- t

4--- r

9---

2x3

------x2

6----- 5

7--- x

4---

WWORKEDORKEDEExample

2

WWORKEDORKEDEExample

3 SkillSH

EET 3.1

mmultiple choiceultiple choice

27x

------

13y

------

p4---

n10------

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84 M a t h s Q u e s t 9 f o r V i c t o r i a

Worded questionsAn important skill in algebra is to convert worded questions, or sentences, intoalgebraic expressions. Worded questions need to be read carefully so that you candecide where to place the pronumerals, coefficients and constants in an expression.

Finding pronumerals, coefficients and constantsThe first step in converting a worded question into an algebraic expression is to identifyany unknown quantities. We then identify the coefficients, constants and the arithmeticoperations that connect them to form an algebraic expression.

In other examples we may need to assign our own pronumeral to an unknown quantity.We need to explain what any pronumeral we introduce into the question represents. Weshould also try to use a pronumeral that is easily identifiable with what it represents.

Georgia studies 4 more subjects than Henry. How many subjects does Georgia study if: a Henry studies 6 subjectsb Henry studies x subjectsc Henry studies y subjects.

THINK WRITE

a Read the question carefully and check for unknown quantities. All quantities are known.

a

Henry studies 6 subjects. The number of subjects studied by Georgia is 4 more than Henry.

Number of subjects studied by Henry = 6. Number of subjects studied by Georgia

= 6 + 4= 10

b Read the question carefully and check for unknown quantities. The number of subjects studied by Henry is the unknown x.

b

The number of subjects studied by Georgia is 4 more than Henry.

The number of subjects studied by Georgia = x + 4.

c Read the question carefully and check for unknown quantities. The number of subjects studied by Henry is the unknown y.

c

The number of subjects studied by Georgia is 4 more than Henry.

The number of subjects studied by Georgia = y + 4.

1

2

1

2

1

2

4WORKEDExample

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a 85

Convert the following sentences into algebraic expressions. a If it takes 8 minutes to iron a single shirt, how long

would it take to iron all of Alan’s shirts?b Brenda has $5 more than Camillo. How much money

does Brenda have?c In a game of Aussie rules, David kicked 3 more goals

than he kicked behinds. How many points did David score? (1 goal scores 6 points; 1 behind scores 1 point.)

THINK WRITE

a Read the question carefully and check for unknown quantities. The number of Alan’s shirts is unknown.

a

Use a pronumeral for the unknown quantity.

Let n = the number of Alan’s shirts.

The total time taken is the time taken to iron 1 shirt multiplied by the number of shirts.

The total time is 8 × n = 8n.

b Read the question carefully and check for unknown quantities. The amount of money that Camillo and Brenda each have is unknown.

b

Use pronumerals for the unknown quantities.

Let b = the amount of money Brenda has.Let c = the amount of money Camillo has.

To find the amount of money Brenda has we must add $5 to the amount that Camillo has.

b = c + 5

c Read the question carefully and check for unknown quantities. The number of goals and behinds kicked by David is unknown.

c

Use pronumerals for the unknown quantities. We need only 1 pronumeral because there were 3 less behinds kicked than goals.

Let g = the number of goals that David kicked.The number of behinds kicked was g − 3.

One goal is worth 6 points so multiply the number of goals by 6. One behind is worth 1 point.

Number of points from goals = 6 × g= 6g

Number of points from behinds = g − 3Add the points from goals and behinds to find the total points scored.

The number of points scored = 6g + g − 3= 7g − 3.

1

2

3

1

2

3

1

2

3

4

5WORKEDExample

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86 M a t h s Q u e s t 9 f o r V i c t o r i a

After converting a worded question to an algebraic expression, it is possible to see ifyour answer is reasonable by substituting values for the pronumerals. For example inpart a of worked example 5 it takes 8 minutes to iron 1 shirt, so it makes sense that ourexpression should predict that 16 minutes are needed to iron 2 shirts. We can check bysubstituting n = 2 in our expression.If n = 2, 8n = 8 × 2

= 16.

Worded questions

1 Jacqueline studies 5 more subjects than Helena. How many subjects does Jacquelinestudy if:a Helena studies 6 subjects?b Helena studies x subjects?c Helena studies y subjects?

2 Dianne and Angela walk home from school together. Dianne’s home is 2 km furtherfrom school than Angela’s home. How far does Dianne walk if Angela’s home is: a 1.5 km from school?b x km from school?

3 Lisa watched television for 2.5 hours today. How many hours will she watchtomorrow if she watches:a 1.5 hours more than she watched today?b t hours more than she watched today?c y hours less than she watched today?

remember1. The first step in converting a worded question into an algebraic expression is to

identify any unknowns and assign a pronumeral to each.2. Worded questions need to be read carefully so that you can decide where to

place the pronumerals, coefficients and constants in an expression.3. We can check to see if an algebraic expression is reasonable by substituting

values for the pronumerals.

remember

3BWWORKEDORKEDEExample

4

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a 874 Convert the following sentences into algebraic expressions.

a If it takes 10 minutes to iron a single shirt, how long would it take to iron all ofAnthony’s shirts?

b Ross has 30 dollars more than Nick. How much money does Ross have?c In a game of Aussie rules, Luciano kicked 4 more goals than he kicked behinds.

How many points did Luciano score? Remember: 1 goal scores 6 points, 1 behindscores 1 point.

5 Jeff and Chris play football for opposing teams, and Jeff’s team won when the two teamsplayed one another. In AFL rules, a goal scores 6 points and a behind scores 1 point.a How many points did Jeff’s team score if they kicked:

i 14 goals and 10 behinds?ii x goals and y behinds?

b How many points did Chris’s team score if his team kicked:i 10 goals and 6 behinds?ii p goals and q behinds?

c How many points did Jeff’s team win by if:i Chris’s team scored 10 goals and 6 behinds, and Jeff’s team scored 14 goals

and 10 behinds?ii Chris’s team scored p goals and q behinds, and Jeff’s team scored x goals and

y behinds?

6 Yvonne’s mother gives her x dollars for each school subject she passes. If she passesy subjects, how much money does she receive?

7 Roberto orders x cents worth of chips from a fish and chip shop, and divides them upequally between y people. What value does each person receive?

8 Brian buys a bag containing x smarties.a If he divides them

equally between n people, how many does each person receive?

b If he keeps half the smarties for himself and divides the remaining smarties equally between n people, how many does each person receive?

9 A piece of licorice is 30 cm long.a If David cuts d cm off, how much licorice remains?b If David cuts off of the remaining licorice, how much licorice has been cut off?c How much licorice remains now?

WWORKEDORKEDEExample

5

14---

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88 M a t h s Q u e s t 9 f o r V i c t o r i a

10 One quarter of a class of x students play tennis on the weekend. One sixth of theclass play tennis and swim on the weekend.a Write an expression to represent the number of students playing tennis on the

weekend.b Write an expression to

represent the number of students playing tennis and swimming on the weekend.

c How many students only play tennis on the weekend ?

11 During a 24-hour period, Vanessa uses her computer for c hours. Her brother Darrenuses it for of the remaining time. a For how long does Darren use the computer?b For how long do Vanessa and Darren use it altogether during a 24-hour period?

12 Marty had a birthday party last weekend, andinvited n friends. The table at right indicatesthe number of friends at Marty’s party at thespecified times during the evening. a How many people arrived between

7.00 pm and 7.30 pm?b Between which times were the most

friends present at the party?c How many friends were invited but did

not arrive?d How many friends were invited in total?e Between which times did the most friends

arrive?f What assumptions have been made in the

previous answers?g Write a paragraph to describe the presence of Marty’s friends at his party.

17---

Time Number of friends

7.00 pm n – 24

7.30 pm n – 23

8.00 pm n – 8

8.30 pm n – 5

9.00 pm n – 5

9.30 pm n – 7

10.00 pm n – 12

10.30 pm n – 18

11.00 pm n – 24WorkS

HEET 3.1

MA

TH

SQUEST

C H A L L

EN

GE

MA

TH

SQUEST

C H A L L

EN

GE

1 Is it possible to shade 6 of the 9 squares in thisgrid so that no three shaded squares are in astraight line (row, column or diagonal)?

2 Jenna has read the first 127 pages of a book.When she reads 39 more pages, she will haveread half the book. How many pages are inthe book?

3 If a father is now five times as old as his son,how many years ago was the son 2 years oldand the father 34?

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a 89

Like termsWe have already seen that a term such as 3xy, actually means 3 × x × y. In fact, all termsconsist of coefficients and pronumerals multiplied (and divided) together. In Year 7 and8 you were introduced to ‘like terms’. Like terms contain the same pronumeral partsand can be collected (added or subtracted) if they appear in an expression.

Identifying like termsLet us begin with a definition of like terms.

Like terms contain the same pronumeral parts.

When comparing terms, look at the pronumeral parts. If both terms contain the samecombination of pronumerals (letters) they are like terms. For example 4xy and −23xyare like terms. Also 4xy and −23yx are like terms, because the order of multiplication ofthe pronumerals does not affect their value.

The terms 5x2y and 3xy are not like terms, even though they both contain the pro-numerals x and y. This is because they contain a different combination. That is, 5x2ycontains x2y which means x × x × y, whereas 3xy contains xy which means x × y.

A list of like terms for 2abc could include 12abc, –6acb, 500bca, abc. Can you seewhy?

A list of terms that are not like 2abc could include 3ab, 5a2bc, −6cb, 12ab2c. Canyou see why?

Collecting like termsWhen like terms appear in an expression they can be collected (added or subtracted). Inorder to decide whether to add or subtract, we look at the sign of each term. This islocated on the left-hand side of each term in an expression. It is helpful to change theorder of the terms before collecting them. For example, we can write the expression4x + 5y − 3x + 7y as 4x − 3x + 5y + 7y. Notice that the sign on the left-hand side of eachterm stays the same. We can now simplify by subtracting the first 2 terms and addingthe last 2 terms.

4x − 3x + 5y + 7y = x + 12y

23---

For each of the following terms, select those terms listed in brackets that are like terms.a 4y (y, −y, 4x, 4xy, −4y)b 5xy (−5xy, 5x, 5yx, 5xz, −xy)c −6abc (−6bca, −6abd, −6a2bc, −2acb, −2ac2b)d −7q2b3e4 (−7q2b2e2, −6b3e4q2, 6q2e4b3, 7q4b3e4, −7q2b2e2)

THINK WRITEa The pronumeral part of 4y is y.

Check the list for terms with the same pronumeral part.a Like terms: y, −y, −4y

b The pronumeral part of 5xy is xy. Check the list for terms with the same pronumeral part.

b Like terms: −5xy, 5yx, −xy

c The pronumeral part of −6abc is abc. Check the list for terms with the same pronumeral part.

c Like terms: −6bca, −2acb

d The pronumeral part of −7q2b3e4 is q2b3e4. Check the list for terms with the same pronumeral part.

d Like terms: −6b3e4q2, 6q2e4b3

6WORKEDExample

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90 M a t h s Q u e s t 9 f o r V i c t o r i a

In the following worked example you will need to look carefully to identify the like terms.

Simplify the following expressions by collecting like terms.a 6b + 5b b 6x + 5y − 4x + 2yc 7ax + 7x − 5a − 6ax d 9a2b − 3ab2 + 2ab

THINK WRITE

a Write the expression. a 6b + 5bIdentify the like terms and simplify. = 11b

b Write the expression. b 6x + 5y − 4x + 2yIdentify the like terms and change the order. = 6x − 4x + 5y + 2ySimplify by collecting like terms. = 2x + 7y

c Write the expression. c 7ax + 7x − 5a − 6axIdentify the like terms and change the order. = 7ax − 6ax + 7x − 5aSimplify by collecting like terms. = ax + 7x − 5a

d Write the expression. d 9a2b − 3ab2 + 2abIdentify the like terms. There are none! Cannot be simplified.

1

2

1

2

3

1

2

3

1

2

7WORKEDExample

Simplify the following expressions.a 6a2 + 9b + 7b2 − 5b b 12 − 4a2b + 2 − 2ba2 c 8ab + 2a2b + 5a2b − ab

THINK WRITE

a Write the expression. a 6a2 + 9b + 7b2 − 5b

Identify the like terms and change the order. = 6a2 + 9b − 5b + 7b2

Simplify by collecting like terms. = 6a2 + 4b + 7b2

b Write the expression. b 12 − 4a2b + 2 − 2ba2

Identify the like terms and change the order. = 12 + 2 − 4a2b − 2ba2

Simplify by collecting like terms. = 14 − 6a2b

c Write the expression. c 8ab + 2a2b + 5a2b − ab

Identify the like terms and change the order. = 8ab − ab + 2a2b + 5a2b

Simplify by collecting like terms. = 7ab + 7a2b

1

2

3

1

2

3

1

2

3

8WORKEDExample

remember1. Like terms contain the same pronumeral parts.2. When like terms appear in an expression they can be collected (added or

subtracted).3. It is helpful to change the order of the terms in an expression before collecting

them, but be careful that the sign on the left of each term stays the same.

remember

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a 91

Like terms

1 For each of the following terms, select those terms listed in brackets that are like terms.a 6ab (7a, 8b, 9ab, −ab, 4a2b2)b −x (3xy, −xy, 4x, 4y, −yx)c 3az (3ay, −3za, −az, 3z2a, 3a2z)d x2 (2x, 2x2, 2x3, −2x, −x2)e −2x2y (xy, −2xy, −2xy2, −2x2y, −2x2y2)f 3x2y5 (3xy, 3x5y2, 3x4y3, −x2y5, −3x2y5)g 5x2p3w5 (−5x3w5p3, p3x2w5, 5xp3w5, −5x2p3w5, w5p2x3)h −x2y5z4 (−xy5, −y2z5x4, −x + y + z, 4y5z4x2, −2x2z4y5)

2 Simplify the following expressions by collecting like terms.a 5x + 2x b 3y + 8y c 7m + 12md 13q − 2q e 17r − 9r f x + 4xg a + 7a h m + 3m i h + 9hj 10v − v k 13a − 9a l 17p − 4pm 5a + 2a + a n 9y + 2y − 3y o 7x − 2x + 8xp 14p − 3p + 5p

3 Simplify the following expressions.a 3m2 + 9m2 b 2q2 + 7q2 c 5x2 − 2x2

d 9p2 − 3p2 e 8m2 − 7m2 f 14w2 − 13w2

g 9x2 − x2 h 5m2 − m2 i 2r2 − r2

4 Simplify the following expressions by collecting like terms.a 6x2 + 2x2 − 3y b 3m2 + 2n − m2 c 9x2 + x − 2x2

d 11xy + 8xy − x2 e 2x2 + 5x − 3x + 7 f 4m2 + 9m − 2m + 3g 6y2 − 2y − 5y + 6 h 7p2 − 6p − 7p + 1 i b2 − 3b + 5b − 1j b2 − 6b + 8b − 5 k c2 − 5c + 2c + 4 l v2 − 7v + 6v − 3m 9h2 − 2h + 3h + 9 n −2g2 − 4g + 5g − 12 o −5m2 + 5m − 4m + 15p −8j2 + 7j − 8j − 1 q −9k2 − 2k + k + 10

5 Simplify the following expressions.a 12a2 + 3b + 4b2 − 2b b 6m + 2n2 − 3m + 5n2 c 3xy + 2y2 + 9yxd 5x2 + 7xy − 2yx e 13m + 9 + 3m − 3 f 11 − 3a2b + 4 − 7ba2

g 3x + 4xy − 2x + 7xy h 13x2 + 5x2y − 9x2 i 9a2b + 2ba2 − 3b2aj 3ab + 3a2b + 2a2b − ab k 9x2y − 3xy + 7yx2 l 4m2n + 3n − 3m2n + 8nm 3x2 − 8x2 + 2 n 5xy + 9x2 − 8yx o 11m2n − 3nm2 + 5mn2

p 6xy + 7x2y − 3x2y − 9xy

6What do the following expressions equal?a 18p − 19p

b 5x2 − 8x + 6x − 9

c 12a − a + 15b − 14b

d −7m2n + 5m2 + 3 − m2 + 2m2n

A p B −p C p2 D −p2 E −1

A 3x − 9 B 3x2 − 9 C 5x2 + 2x − 9 D 5x2 − 2x − 9 E 5x2 − 11

A 11a + b B 12 C 11a − b D 13a + b E 13a − b

A −9m2n + 4m2 + 3 B −9m2n + 8 C −5m2n + 8D −5m2n − 4m2 + 3 E −5m2n + 4m2 + 3

3CWWORKEDORKEDEExample

6

WWORKEDORKEDEExample

7a

Mathcad

Collectinglike terms

SkillSH

EET 3.2WWORKEDORKEDEExample

7b, c, d

WWORKEDORKEDEExample

8

mmultiple choiceultiple choice

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92 M a t h s Q u e s t 9 f o r V i c t o r i a

Algebra rectanglesThe figure drawn at right is a rectangle. For the purpose of this activity we are going to call this rectangle an ‘algebra’ rectangle.

The length of each ‘algebra’ rectangle is x cm and the width is 1 cm.These ‘algebra’ rectangles are put together to form larger rectangles in one of

two ways.

Long algebra rectangle Tall algebra rectangle

Let’s find the perimeter of each ‘algebra’ rectangle.Perimeter of 3-long ‘algebra’ rectangle; P = (x + x + x + x + x + x) + (1 + 1)

= 6x + 2Perimeter of 3-tall ‘algebra’ rectangle; P = (x + x) + (1 + 1 + 1 + 1 + 1 + 1)

= 2x + 61 Find the perimeter of each of the ‘algebra’ rectangles and put your results into

the table below.

2 Can you see a pattern? What would be the perimeter of a 20-long algebra rectangle and a 20-tall algebra rectangle?

3 What type of algebra rectangle will have the greater perimeter, a long algebra rectangle or a tall algebra rectangle?

4 Does the answer to question 3 depend on the value of x?5 Suppose now that I use n tiles to make an n-tile-long algebra rectangle and an

n-tile-tall algebra rectangle. Write an expression for the perimeter of each.

x1

xxx

xxx11

x

x

111

111

Type of ‘algebra’ rectangle Perimeter

Type of ‘algebra’ rectangle Perimeter

1-long 1-tall

2-long 2-tall

3-long 6x + 2 3-tall 2x + 6

4-long 4-tall

5-long 5-tall

6-long 6-tall

7-long 7-tall

8-long 8-tall

9-long 9-tall

10-long 10-tall

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a

93

This medical firstThis medical first occur occurrred in 1ed in 1953!953!

7a

6a

4a + 7b 5a + ab 5b + ab 3b + 2ab 4b + 3ab 8a + 2ab

7b a + 2ab 3a + b –a + 4ab –a – b 4ab –2b + 4ab 4a + ab 9a +6b

–4b 5a – b 2b –3a 8a b 8b a + b 8ab –2a –ab 10b a 6ab

8a – b –2ab 5ab ab 2a 7ab 3b 4a 2a + 7ab b – ab

7a + 5ab

5a + 3a = 2ab + 5b – ab =

b – 2b – 3b =10a + a – 7a = ab + 8ab – 2ab =

4a + 3b – 3a – 2b =5a + 4b – a + 3b = 12b – b – 4b =

5a – b =16a – 9a = 2b + 13b – 4b – 3b =

7b + 3a – 6b = 3ab + b – ab + 2b = 2ab + ab + 2ab =

2a + 3a + a = 7a + 2b + 8b – 7a = 5a + 4ab – 6a =

3ab + 5a – 2ab = ab + a + 3ab – a = 7a + a – 10a – a =

a + ab + 7a + ab = 7ab + 4a + 3a – 2ab = 4a + 8ab – 2a – ab =

6ab – 3ab – 2ab = b + 3b + 7ab – 4ab = 3ab – ab – 3ab =

3a + 2b – 2a – a = 6a + 2a + 3b – 4b = 4b + ab – 3b – 2ab =

10ab – 8ab + a = –3a + a + 4a = 3a + b – 2a – a =

b + 6b – 4b = 5ab + 4a – 4ab = 4a – 4b – 5a + 3b =

7ab – 4ab + 3ab = 12ab – 7ab + 3ab = –7a + 2a – a + 4a =

6a + 4b + 3a + 2b = 2a – 5a + 4a =

3ab – 5ab =

2ab + b + 2ab – 3b =

MQ9 Vic ch 03 Page 93 Wednesday, September 19, 2001 11:07 AM

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94 M a t h s Q u e s t 9 f o r V i c t o r i a

1 For the expression 10xy + 12y2 − 9 + 4z state the largest coefficient.

2 Write an algebraic expression for a number 3 more than x.

3 If there are 550 people and y of them don’t vote, write an expression for the numberwho do vote.

4 Karlie has four birds. How many will she have if she buys t more?

5 Gary has a piece of material 100 cm long. If he cuts q cm off, how long is theremaining piece?

6Jessie sells homemade lemonade at p cents per glass and then sells it to q people. Thetotal amount of money, in cents, made by Jessie is:

7 One third of a class of x students watch football only and one fifth watch soccer only.Write an algebraic expression to represent the number of students who watch footballor soccer.

8 Simplify the expression −4d − 6d.

9 Simplify 10x2 + 7x − 5x2.

10 Simplify 9x2 + 4y + 3x2 − 2y.

Multiplication and divisionWhen multiplying and dividing algebraic terms, it is not necessary to have like terms.In fact any terms can be multiplied or divided and the result is a single new term.

Multiplication Consider the product 8x × 3y which equals 24xy.

When we multiply the terms together we can rearrange the product and consider themultiplication of the coefficients (number parts) separately. This is because,

8x × 3y = 8 × x × 3 × y= 8 × 3 × x × y (since order is not important) = 24 × x × y = 24xy (The × signs still exist but are not shown.)

When a pronumeral is multiplied by itself, we can use a power or index rather thanwriting the pronumeral each time. For example,

8x × 2xy × 3x = 8 × 2 × 3 × x × x × y × x = 48x3y

You will learn more about powers and indices later in this book.

A p + q B pq C D p − q E

1

mmultiple choiceultiple choice

pq--- q

p---

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a 95

DivisionWhen dividing terms, we write the division as a fraction and try to simplify by cancel-ling the numerator and denominator by any common factors. The coefficients (numberparts) and pronumeral parts can be treated separately.

For example, 6xy ÷ 3x would be written as:

=

=

= 2y

Simplify the following.a 4a × 3a b 4a × 2b × ac 5x × 3y × −2x d 7ax × −6bx × −2abx

THINK WRITE

a Write the algebraic terms. a 4a × 3aRearrange, writing the coefficients first.

= 4 × 3 × a × a

Multiply the coefficients and pronumerals separately.

= 12 × a × a= 12 × a2

= 12a2

b Write the algebraic terms. b 4a × 2b × aRearrange, writing the coefficients first.

= 4 × 2 × a × a × b

Multiply the coefficients and pronumerals separately.

= 8 × a2 × b= 8a2b

c Write the algebraic terms. c 5x × 3y × −2xRearrange, writing the coefficients first.

= 5 × 3 × −2 × x × x × y

Multiply the coefficients and pronumerals separately.

= −30 × x2 × y= –30x2y

d Write the algebraic terms. d 7ax × −6bx × −2abxRearrange, writing the coefficients first.

= 7 × −6 × −2 × a × a × x × x × x × b × b

Multiply the coefficients and pronumerals separately. The simplified term is often written with the pronumerals in alphabetical order.

= 84 × a2 × x3 × b2

= 84a2b2x3

1

2

3

1

2

3

1

2

3

1

2

3

9WORKEDExample

6xy3x

---------

6 x y××3 x×

---------------------

2 y×1

------------

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96 M a t h s Q u e s t 9 f o r V i c t o r i a

Simplify the following.

a b c 8ab ÷ 16a2b d −5xyz ÷ 9x2

THINK WRITE

a Write the term. a

Cancel 6 and 3 (common factor of 3). =

= 2x

b Write the term. b

Cancel 4 and 10 (common factor 2). Cancel y from numerator and denominator.

=

c Write the terms and express as a fraction. The term a2 means aa.

c 8ab ÷ 16a2b

=

=

Cancel 8 and 16 (common factor 8). Cancel a and b from numerator and denominator.

=

d Write the terms and express as a fraction. The term x2 means xx.

d −5xyz ÷ 9x2

=

=

We cannot cancel 5 and 9. Cancel x from numerator and denominator.

=

6x3

------ 4xy10 yz------------

16x3

------

22x1

------

14xy10yz-----------

22x5z------

1

8ab16a2b---------------

8ab16aab----------------

21

2a------

1

5xyz9x2

-----------–

5xyz9xx-----------–

25yz9x--------–

10WORKEDExample

remember1. When multiplying and dividing algebraic terms it is not necessary to have like

terms.

2. For multiplication we can multiply the coefficients (number parts) and the pronumeral parts separately.

3. A division problem should be expressed as a fraction.

4. For division, try to simplify by cancelling the numerator and denominator by any common factors.

remember

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a 97

Multiplication and division

1 Simplify the following.a 3m × 2n b 4x × 5y c 2p × 4qd 5x × −2y e 3y × −4x f −3m × −5ng 5a × 2a h 4y × 5y i 5p × pj m × 7m k 3mn × 2p l −6ab × b m −5m × −2mn n −6a × 3ab o −3xy × −5xy × 2x p 4pq × −p × 3q q 4c × −7cd × 2c r −3a × −5ab × 2ab

2 Simplify the following.

a b c

d e 12m ÷ 3 f 14x ÷ 7

g −21x ÷ 3 h −32m ÷ 8 i

j k l

m n o

p q r −7xy2z2 ÷ 11xyz

3 Simplify the following.a 5x × 4y × 2xy b 7xy × 4ax × 2y c x × 4xy × 3yx

d e f

g −4a × −5ab2 × 2a h −a × 4ab × 2ba × b i 2a × 2a × 2a × 2a

3DWWORKEDORKEDEExample

9 SkillSH

EET 3.3

Mathcad

Multiplicationand division

WWORKEDORKEDEExample

10 6x2

------ 9m3

------- 12y6

---------

8m2

-------

4m8

-------

6x18------ 8mn

18n----------- 16xy

12y------------

210m---------- 6ab

12a2b--------------- 28xyz

14x--------------

70ab2

4b--------------- 2x2yz

8xz--------------

6x2y12y2----------- 15x2ab–

12b2x2--------------------- 2 p3q2

p3q2--------------- Wo

rkSHEET 3.2

MA

TH

SQUEST

C H A L L

EN

GE

MA

TH

SQUEST

C H A L L

EN

GE

1 Place two different sets of 2 coloured counters(or coins) on the grid as shown. Show how youcan swap the position of the two sets of coun-ters in exactly 8 moves. A counter can slide into an empty square nextto it or can jump over another counter into an empty space. Recordyour solution.

2 Try this game again but with twodifferent sets of 3 counters and a gridof 7 squares. Try to swap the two setsof counters in exactly 15 moves. Record your solution.

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= =

= =

= =

= =

= =

= =

=

=

=

= =

= =

= =

9c 2 5ce 2 4be

6ab

6c 4

3ac 2

42be 24ae 4ac 2a 7b15a 2 25c 2 16a 2 7ce 2 12a 2b12b 2 6c 3 a 2

5ac 26ce 2 10e 2 14b 3 6e 3 8ab8e 335ae 5e 2c 3a3ae 5ab

2ac

4a 21ae5a 2

5b

10a 2c 2

2a

50a 2b 2

10ab

24a 2b 2

8ab 2

18c 3a 2

2ca 2

64b 3e 2

16b 2e

35c 2e 2

5c

a 2bb

15a 2e 5a

16bc 8b

10abc 2ac

= 3a x 5a = = 5a x 7e =

= 2b x 3a =

= 2b x 3b x 2 =

= 7b x 2b x b =

= 10ae ÷ 5e =

= 30ae ÷ 6a =

= 20a 2e ÷ 4e =

= 12a 2c ÷ 6a =

= 100c 2e 2 ÷ 20c

= 21b 2e ÷ 3be

= 2c 2 x 3c 2 =

= a x 8a ÷ 2a =

=

= 32a 2b ÷ 2b =

= 2c x c x 3c =

= 2b x 3e x 7 =

= 2a x 4b =

= e x 2e x 3e =

= 5e x 2e =

= 7e x 3a =

= 2c x 2a =

= 3e x 2c x e =

= 6a x 2b x a =

= 2e x 2e x 2e =

= 2 x 3a x 4e = = 5c x 5c =

= c x 3a x c =

98 M a t h s Q u e s t 9 f o r V i c t o r i a

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a 99

Algebraic fractionsAlgebraic fractions contain pronumerals that may represent particular numbers orchanging values.

The methods for dealing with algebraic fractions follow the same principles that weused for numerical fraction questions, such as + or ÷ . That is, find a commondenominator for addition and subtraction, or use reciprocals for division.

Adding and subtracting algebraic fractionsTo add or subtract algebraic fractions we perform the following steps.Step 1 Find the lowest common denominator (LCD) by finding the lowest

common multiple (LCM) of the denominators.Step 2 Rewrite each fraction as an equivalent fraction with this common

denominator.Step 3 Add (or subtract) the new numerators.

25--- 5

7--- 3

4--- 4

5---

Simplify each of the following expressions.

a + b − c +

THINK WRITE

a Write the expression. a +

Find the lowest common denominator (LCD). The lowest common multiple (LCM) of 2 and 5 is 10.

= +

Add the numerators. =

b Write the expression. b −

Find the LCD. The LCM of 3 and 7 is 21. = −

Subtract the numerators. =

c Write the expression. c +

Find the LCD. The LCM of 5 and 6 is 30. = +

Add the numerators. =

Expand the brackets in the numerator. =

Simplify the numerator by collecting like terms. =

x2--- x

5--- y

3--- 2 y

7------ y 1+

5------------ x y+

6------------

1x2--- x

5---

2 5x10------ 2x

10------

37x10------

1y3--- 2y

7------

2 7y21------ 6y

21------

3y

21------

1y 1+

5------------ x y+

6------------

2 6 y 1+( )30

-------------------- 5 x y+( )30

--------------------

36 y 1+( ) 5 x y+( )+

30-----------------------------------------------

46y 6 5x 5y+ + +

30----------------------------------------

5 5x 11y 6+ +30

-------------------------------

11WORKEDExample

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100 M a t h s Q u e s t 9 f o r V i c t o r i a

If pronumerals appear in the denominator we can treat these separately to theircoefficients (numbers).

In such a case the lowest common denominator (LCD) is found by finding the lowestcommon multiple (LCM) of the coefficients, then including in the LCD everypronumeral used.

Multiplying and dividing algebraic fractionsThe rules for multiplication and division are the same as for numerical fractions.

When multiplying algebraic fractions, multiply the numerators and multiply thedenominators, then cancel any common factors in the numerator and denominator.

Simplify the following expressions.

a − b + c −

THINK WRITE

a Write the expression. a −

Find the LCD. The LCM of 1 and 2 is 2. The only pronumeral is x, so include it in the LCD. The LCD is 2x.

= −

Subtract the numerators. = −

b Write the expression. b +

Find the LCD. The LCM of 3 and 5 is 15. The only pronumeral is y so include it in the LCD. The LCD is 15y.

= +

Add the numerators. =

c Write the expression. c −

Find the LCD. The LCM of 5 and 10 is 10. The only pronumeral is z, so include it in the LCD. The LCD is 10z.

= −

Subtract the numerators. =

1x--- 3

2x------ 2x

3 y------ 3

5 y------ 6

5z----- 3x

10------

11x--- 3

2x------

2 22x------ 3

2x------

31

2x------

12x3y------ 3

5y------

2 10x15y--------- 9

15y---------

310x 9+

15y------------------

165z----- 3x

10------

2 1210z-------- 3xz

10z--------

312 3xz–

10z--------------------

12WORKEDExample

MQ9 Vic ch 03 Page 100 Monday, September 17, 2001 9:03 AM

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a 101

When dividing algebraic fractions, change the division sign to a multiplication sign andwrite the following fraction as its reciprocal. This is the same times and tip method thatwas covered in chapter 1 using numerical fractions.

Simplify each of the following.

a × b ×

THINK WRITE

a Write the algebraic fractions. a ×

Multiply the numerators and multiply the denominators. =

Check for common factors in the numerator and denominator and cancel. The numbers 6 and 2 have a common factor of 2.

=

b Write the algebraic fractions. b ×

Multiply the numerators and multiply the denominators. =

Check for common factors in the numerator and denominator and cancel. The numbers 2 and 6 have a common factor of 2. Cancel y.

=

x2--- 6

y--- y

2x------ 2z

3 y------

1x2--- 6

y---

26x2y------

3 3xy

------

1y

2x------ 2z

3y------

22yz6xy---------

3 z3x------

13WORKEDExample

Simplify each of the following.

a ÷ b ÷

THINK WRITE

a Write the algebraic fractions. a ÷

Change the division sign to a multiplication sign and write the second fraction as its reciprocal.

= ×

Multiply the numerators and multiply the denominators. =

Check for common factors in the numerator and denominator. Cancel x.

=

b Write the algebraic fractions. b ÷

Change the division sign to a multiplication sign and write the second fraction as its reciprocal.

= ×

Multiply the numerators and multiply the denominators. =

Check for common factors in the numerator and denominator. Cancel x.

=

1x--- 4

x--- 3xy

2--------- 4x

9 y------

11x--- 4

x---

2 1x--- x

4---

3x

4x------

4 14---

13xy

2--------- 4x

9y------

2 3xy2

--------- 9y4x------

327xy2

8x--------------

4 27y2

8-----------

14WORKEDExample

MQ9 Vic ch 03 Page 101 Monday, September 17, 2001 9:03 AM

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102 M a t h s Q u e s t 9 f o r V i c t o r i a

Algebraic fractions

1 Simplify each of the following expressions.

a + b − c −

d + e + f −

g − h − i +

j −

2 Simplify each of the following expressions.

a − b + c −

d − e − f +

3 Simplify each of the following.

a × b × c ×

d × e × f ×

g × h × i ×

j × k × l ×

m × n × × o × ×

remember1. Algebraic fractions contain pronumerals that may represent particular numbers

or changing values.2. To add or subtract algebraic fractions, we use the same method as with

numerical fractions.(a) Find the lowest common denominator (LCD) by finding the lowest

common multiple (LCM) of the denominators.(b) Rewrite each fraction as an equivalent fraction with this common denominator.(c) Add (or subtract) the new numerators.

3. When multiplying algebraic fractions, multiply the numerators, and multiply the denominators. Cancel any common factors in the numerator and denominator.

4. When dividing algebraic fractions, change the division sign to a multiplication sign and write the following fraction as its reciprocal (swap the numerator with the denominator). Continue as for multiplying algebraic fractions.

remember

3E

SkillSH

EET 3.4 WWORKEDORKEDEExample

11

SkillSH

EET 3.5

x3--- x

4--- y

2--- y

3--- m

8---- m

4----

x6--- x

12------ m

2---- m

7---- t

3--- t

5---

3a2

------ a5--- 2 p

3------ 5 p

6------ 4q

5------ q

3---

5x6

------ 2x3

------

WWORKEDORKEDEExample

12

Mathca

d

Algebraicfractions

32 p------ 2

p--- 1

2x------ 3

5x------ 3

4m------- 7

2m-------

85b------ 5

4b------ 11

6c------ 4

9c------ 2

3y------ 4

5y------

SkillSH

EET 3.6 WWORKEDORKEDEExample

13 23--- 9

2--- 3

5--- 10

3------ 5

12------ 3

5---

415------ 3

4--- x

3--- 9

x--- 4

y--- y

12------

43--- m

16------ n

9--- 3

2--- 7m

5------- 10

m------

53x------ x

15------ 20

3y------ 6

5--- 2x

3------ 15

6------

4m27------- 9

7m------- 2

15------ 7

p--- p

21------ x

22------ 11

12------ 6

x---

MQ9 Vic ch 03 Page 102 Monday, September 17, 2001 9:03 AM

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a 1034 Simplify each of the following.

a ÷ b ÷ c ÷

d ÷ e ÷ f ÷

g ÷ h ÷ i ÷

j ÷ k ÷ l ÷

m ÷ n × ÷ o × ÷

1 Write down an algebraic expression for the sum of 5 times m and 4 times n.

2 Ben has 6 kittens. How many will he have if he sells x of them?

3 Karen has a bread stick 60 cm long. If she cuts off p cm, how much bread remains?

4 Simplify 4y2 − 7y + 5y + 2.

5 Simplify −2m × −4n.

6 Simplify .

7 Simplify + .

8 Simplify × .

9 Simplify + .

10 Simplify ÷ .

Substitution and formulasIn mathematics, science and engineering, algebraic expressions and formulas arecommonly used. For example, in chapter 2 you learned the formula for Pythagoras’theorem (c2 = a2 + b2) which enabled you to find the unknown side in a right-angledtriangle. In this section we will look at how to substitute particular values for thepronumerals in an expression or formula.

SkillSH

EET 3.7WWORKEDORKEDEExample

14 25--- 2

15------ 3

4--- 3

8--- 5

6--- 15

6------

910------ 36

10------ x

3--- x

9--- 4

m---- 12

m------

a5--- a

20------ 6

b--- 20

b------ 3a

14------ a

7---

214

------ 3b--- 6m

15------- 2

3--- ab

9------ a

24------

2m3 p------- 10m

9 pq---------- 3

5--- 10

m------ 12

m------ 3x

8------ 2y

15------ y

4---

GAMEtime

Algebra— 001

2

8x64------

x2--- x

5---

xy5----- 30z

x--------

34b------ 2

5b------

2x3y------ 24x

21yz-----------

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104

M a t h s Q u e s t 9 f o r V i c t o r i a

Substitution

We can evaluate (find the value of) an algebraic expression if we replace the pro-numerals with their known values. This process is called substitution. Consider theexpression 4

x

+

3

y

. If we substitute the known values

x

=

2 and

y

=

5,we obtain 4

×

2

+

3

×

5

=

8

+

15

=

23Rather than showing the multiplication signs, it is common in mathematics to write

the substituted values in brackets. We would write the example above as:4

x

+

3

y

=

4(2)

+

3(5)

=

8

+

15

=

23

Substitution into formulas

A formula expresses one quantity in terms of oneor more other quantities. For example, the formulafor the area of a rectangle is given by:

Area

=

length

×

width or

A

=

l

×

w.

If a particular type of kitchen tile has a length,

l

=

20 cm, and width,

w

=

15 cm, we can substi-tute these values into the formula to find its area.

A

=

l

×

w

=

20

×

15

=

300 cm

2

The same formula can be used to calculate thearea of a tile of different size (provided it is rec-tangular in shape) by substituting whatever thelength (

l

) and width (

w

) happen to be.

If x = 3 and y = −2, evaluate the following expressions.a 3x + 2y b 5xy − 3x + 1 c x2 + y2

THINK WRITEa Write the expression. a 3x + 2y

Substitute x = 3 and y = −2. = 3(3) + 2(−2)Evaluate. = 9 − 4

= 5b Write the expression. b 5xy − 3x + 1

Substitute x = 3 and y = −2. = 5(3)(−2) − 3(3) + 1Evaluate. = −30 − 9 + 1

= −38c Write the expression. c x2 + y2

Substitute x = 3 and y = −2. = (3)2 + (−2)2

Evaluate. = 9 + 4= 13

1

2

3

1

2

3

1

2

3

15WORKEDExample

20 cm15 cm

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a

105

Ivan, the electrician, knows that the formula for the voltage in an electrical circuit can be found using the formula known as Ohm’s Law: V = IR where I = current in amperes, R = resistance in ohms and V = voltage in volts. Find V when:a I = 2 amperes, R = 10 ohmsb I = 20 amperes, R = 10 ohmsc I = 0.6 amperes, R = 6600 ohms.

THINK WRITEa Write the formula. a V = IR

Substitute I = 2 and R = 10. = (2)(10)Evaluate and express the answer in the correct units.

= 20 volts

b Write the formula. b V = IRSubstitute I = 20 and R = 10. = (20)(10)Evaluate and express the answer in the correct units.

= 200 volts

c Write the formula. c V = IRSubstitute I = 0.6 and R = 6600. = (0.6)(6600)Evaluate and express the answer in the correct units.

= 3960 volts

1

2

3

1

2

3

1

2

3

16WORKEDExample

The distance (x) travelled by an object is given by the formula:x = ut + at2, where u = the starting speed in m/s, a = acceleration in m/s2 and t = time in seconds.

A car starts at a speed of 50 m/s and accelerates at 6 m/s2 for 7 s. How far has the car travelled?THINK WRITE

Write down the formula. x = ut + at2

Substitute u = 50, a = 6, and t = 7. = (50)(7) + (6)(7)2

Evaluate and express the answer in the correct units. = 350 + 147= 497 m

12---

112---

212---

3

17WORKEDExample

remember1. We can evaluate (find the value of) an algebraic expression if we substitute the

pronumerals with their known values. 2. Rather than showing the multiplication signs, it is common in mathematics to

write the substituted values in brackets.3. Formulas can be used to calculate quantities when known values are substituted.

remember

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106 M a t h s Q u e s t 9 f o r V i c t o r i a

Substitution and formulas

1 If x = 4 and y = −3, evaluate the following expressions.a 4x + 3y b 3xy − 2x + 4 c x2 − y2

2 The formula for the voltage in an electrical circuit can be found using the formulaknown as Ohm’s Law: V = I Rwhere I = current in amperes, R = resistance in ohms and V = voltage in volts. Find V when:a I = 4 amperes, R = 8 ohms b I = 25 amperes, R = 10 ohmsc I = 0.8 amperes, R = 300 ohms d I = 3.5 amperes, R = 70 ohms

3 Evaluate each of the following by substituting the given values into each formula. a If A = bh, find A when b = 5 and h = 3.

b If d = find d when m = 30 and v = 3.

c If A = xy, find A when x = 18 and y = 2.

d If A = (a + b)h, find A when h = 10, a = 7 and b = 2.

e If V = , find V when A = 9 and H = 10.

f If v = u + at, find v when u = 4, a = 3.2 and t = 2.1.g If t = a + (n − 1)d, find t when a = 3, n = 10 and d = 2.

h If A = (x + y)h, find A when x = 5, y = 9 and h = 3.2.

i If A = 2b2, find A when b = 5.j If y = 5x2 − 9, find y when x = 6.k If y = x2 − 2x + 4, find y when x = 2.l If a = −3b2 + 5b − 2, find a when b = 4.m If s = ut + at2 find s when u = 0.8, t = 5 and a = 2.3.

n If F = find F when m = 6.9, p = 8 and r = 1.2 (answer to

2 decimal places).o If C = d, find C when = 3.14 and d = 11.

4 a The area of a triangle is given by the formula A = bh, where b is the length of the base and h is the height of the triangle. Find A when:

i b = 6 cm and h = 4 cmii b = 5 cm and h = 2 cmiii b = 3 cm and h = 1 cm.

b The formula to convert degrees Fahrenheit (F) to degrees Celsius (C) is

C = (F − 32). Find C when:

i F = 59 ii F = 44 iii F = 32.c The length of the hypotenuse of a

right-angled triangle (c) can be found using the formula, , where a andb are the lengths of the other two sides. Find c when:

i a = 3 and b = 4 ii a = 12 and b = 5 iii a = 8 and b = 6.

3F

SkillSH

EET 3.8 WWORKEDORKEDEExample

15

WWORKEDORKEDEExample

16

EXCEL

Spreadsheet

Substitution

Mathca

d

Substitutionmv----

EXCEL

Spreadsheet

Substitutiongame

12---

12---

AH3

--------

12---

12---

mpr2

-------

π πWWORKEDORKEDEExample

17

12---

59---

c a2 b2+=

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a 107d If the volume of a prism (V) is given by

the formula V = Ah, where A is the area of the cross-section and h is the height of the prism, find V when:

i A = 7 cm2 and h = 9 cmii A = 10 cm2 and h = 58 cmiii A = 3.6 cm2 and h = 2.3 cm.

e Find the number of edges (E) on a prism if it can be found using E = F + V − 2 where F is the number of faces and V is the number of vertices when:

i F = 5 and V = 7ii F = 7 and V = 10iii F = 10 and V = 12.

f The kinetic energy (E) of an object is found by using the formula E = mv2 where m is the mass and v is the velocity of the object. Find E when:

i m = 3 and v = 3.6ii m = 5 and v = 3.6iii m = 0.2 and v = 10.

g The volume of a cylinder (v) is given by v = r2h where r is the radius and h is theheight of the cylinder. Find v if = 3.14 and:

i r = 7 and h = 3ii r = 49 and h = 9.2 (to 2 decimal places)iii r = 2.5 and h = 3.98 (to 2 decimal places).

h The surface area of a cylinder (S) is given by S = 2πr (r + h). Find S (to 2 decimalplaces) if r is the radius of the circular end, h is the height of the cylinder and

= 3.14, when:i r = 14 and h = 5 ii r = 2 and h = 10iii r = 3.4 and h = 7.2.

Good healthAt the beginning of this chapter we considered the case of Richard who is 86 kg and 1.75 m tall. We know that the Body Mass Index (B) can be found using the

formula ,

where m is the person’s mass, in kilograms, and h is the height, in metres.1 Calculate Richard’s Body Mass Index.2 A person is considered to be in a healthy weight range if 21 ≤ B ≤ 25. Comment

on Richard’s weight for a person of his height.3 Calculate the Body Mass Index for each of the following people:

a Judy who is 1.65 metres tall and has a mass of 52 kilogramsb Karen who is 1.78 metres tall and has a mass of 79 kilogramsc Manuel who is 1.72 metres tall and has a mass of 85 kilograms.

4 Calculate your own Body Mass Index.

12---

ππ

GAMEtime

Algebra— 002

WorkS

HEET 3.3π

Bmh2-----=

MQ9 Vic ch 03 Page 107 Monday, September 17, 2001 9:03 AM

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108

M a t h s Q u e s t 9 f o r V i c t o r i a

H E R O N ( c . 1 0 0 A D ) a n dB R A H M AG U P TA ( 5 9 8 – 6 6 8 A D )

Heron and Brahmagupta were mathematicians who developed formulas for finding the area of geometrical shapes.

HeronHeron, also called Hero, was a Greek mathematician and inventor. He lived in Alexandria in Egypt in the first century AD. Translations of his works still survive. His interests included numbers, geometry, astronomy and mechanics. He invented many machines and toys such as fountains, syphons and steam engines which were operated by water, steam or compressed air.

In geometry his work included methods of finding the areas of regular shapes and polygons. He is credited with the discovery of a formula for finding the area of a triangle. Heron’s formula states that for any triangle with sides a, b and c then the area of the triangle is

where s = .

Heron’s formula can be used to prove Pythagoras’ theorem.

BrahmaguptaBrahmagupta was a mathematician and astronomer who lived during the seventh century AD in north west India. He wrote

several books but his most important work was called Brahma-sphuta-siddhanta which means ‘the opening of the universe’. It described the Hindu astronomical system. Two of its 25 chapters are devoted to mathematics. They include work on arithmetic progressions, quadrilaterals, right-angled triangles, volumes and surfaces. Brahmagupta

developed a formula for the maximum area of a quadrilateral where for any quadrilateral of side lengths a, b, c and d the maximum area is

where s = .

It is interesting to note that this is the same as Heron’s formula with one of the four side lengths set to zero.

A general case for a quadrilateralA general formula for the area of any quadrilateral is

where s = and A and C are the

internal angles, or

where a, b, c and d are the side lengths and p and q are the diagonal lengths.

Questions1. Name one of Heron’s inventions.2. What does Heron’s formula determine?3. What was Brahmagupta’s job?4. What does Brahmagupta’s formula determine?5. What is the connection between the formulas

of Heron and Brahmagupta?

0 500 1000 km

N

AlexandriaHeron lived here.

Turkey

Greece

Baghdad

Brahmaguptalived here.

s s a–( ) s b–( ) s c–( )

a b c+ +2

---------------------a b

c

s s a–( ) s b–( ) s c–( ) s d–( )a b c d+ + +

2-------------------------------

a

bd

c

s a–( ) s b–( ) s c–( ) s d–( ) abcd cosA C+

2--------------

a b c d+ + +2

------------------------------

14--- 4 p2q2 (b2– d2 a2– c2 )

2–+

History of mathematics

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C h a p t e r 3 I n t r o d u c t o r y a l g e b r a 109

Copy the sentences below. Fill in the gaps by choosing the correct word or expression from the word list that follows.

1 A is a group of letters and numbers within an expression.2 The number in the term −6xy is called a .3 The letter in the term −7a is called a .4 An is an algebraic expression that contains an equals sign.5 7abc and 6bca are called terms.6 The expression 7x + 5y + 3x − 6y is equivalent to .7 The first step in converting a into an algebraic expression is

to identify any pronumerals.8 When appear in an expression they can be collected (added

or subtracted).9 When adding or subtracting algebraic fractions, you first must find the

.10 When multiplying algebraic fractions, multiply the numerators together

and the together. Try to simplify by the numer-ator and denominator by any common factors.

11 When dividing algebraic fractions, change the division sign to a sign and write the following fraction as its .

12 We can evaluate an algebraic expression if we the pro-numerals with their known values.

summary

W O R D L I S Tcancellingworded questioncoefficientterm

equationreciprocal10x − ylike terms

likepronumeralsubstituteLCD

multiplicationdenominators

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110 M a t h s Q u e s t 9 f o r V i c t o r i a

1 a For the expression −8xy2 + 2x + 8y2 − 5:i state the number of terms ii state the coefficient of the first termiii state the constant termiv state the term with the smallest coefficient.

b Write expressions for the following, where x and y represent numbers:i a number 8 more than yii the difference between x and yiii the sum of x and yiv 7 times the product of x and yv 2 times x is subtracted from 5 times y.

2 a Leo receives x dollars for each car he washes.If he washes y cars, how much money does he earn?

b A piece of rope is 24 metres long.i If George cuts k m off, how much is left?ii After George has cut k m off, he divides the rest into

three pieces the same length. How long is each piece?

3 Simplify the following expressions by collecting like terms.

a 8p + 9p b 7m2 + 2m2 + m2 c 5y2 + 2y − 4y2

d 8ab + 3b2 + 2ba e 9s2t − 12s2t f 5x + 6xy − x + 3xy

g 11c2d − 2cd + 5dc2 h 7x2y − 8 − 2x2y + 2 i n2 − p2q − 3p2q + 6

j 8ab + 2a2b2 − 5a2b2 + 7ab

4 Simplify the following expressions.a 6a × 2b b −5y × y c 2ab × b d −2p × −3pq e 2xy × 4yx

f g h i 18 ÷ 4b j 3a ÷ 15a

5 Simplify the following expressions.

a + b − c − d + e × f ×

g × h × i × × j ÷

k ÷ l ÷ m ÷ n ÷

6 If y = 5x2 + 2x − 1, find y when:a x = 2 b x = 5.

7 The volume (v) of each of these paint tins is given by the formula, v = πr2h, where r is the radius and h is the height of the cylinder. Find v (to 2 decimal places) if π = 3.14.a r = 7 cm and h = 2 cm b r = 9.3 cm and h = 19.8 cm.

3A

CHAPTERreview

3B

3C

3D10a

2--------- 4x

12------ 2

14m----------

3Ex2--- x

5--- x

6--- x

12------ 3x

4------ 5x

8------ 5

12m---------- 7

8m------- a

6--- 12

a------ 2

y--- y

8---

72x------ x

14------ 1

3 p------ 6 p

5------ a

12------ 4

5--- 10

a------ a

2--- a

4---

5m---- 30

m------ 24

7y------ 8

y--- 5m

6y------- 15m

3xy---------- 20a

14c--------- 4ab

7c---------

3F

3Ftesttest

CHAPTERyyourselfourself

testyyourselfourself

3

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FAQs

How do you simplify algebraic expressions? ›

Simplifying an algebraic expression means writing the expression in the most basic way possible by eliminating parentheses and combining like terms. For example, to simplify 3x + 6x + 9x, add the like terms: 3x + 6x + 9x = 18x.

How do you substitute values into an algebraic expression? ›

To substitute a number into an algebraic expression, all you need to do is re-write the expression in exactly the same way, except replacing the variable (letter) with the number. It always makes it clearer to put the number in brackets too. Then you can simplify your new expression and you have your answer!

What is the concept of Pronumerals as a way of representing numbers? ›

A pronumeral is a letter that stands for a number. If a pronumeral could represent any number rather than just one, it is also called a variable. a × b is written as ab and a ÷ b is written as a b . A term consists of numbers and pronumerals combined with multiplication or division.

How do I simplify the expression? ›

Simplifying expressions mean rewriting the same algebraic expression with no like terms and in a compact manner. To simplify expressions, we combine all the like terms and solve all the given brackets, if any, and then in the simplified expression, we will be only left with unlike terms that cannot be reduced further.

How to evaluate algebraic expressions? ›

To evaluate an algebraic expression, you have to substitute a number for each variable and perform the arithmetic operations. In the example above, the variable x is equal to 6 since 6 + 6 = 12. If we know the value of our variables, we can replace the variables with their values and then evaluate the expression.

How to find an algebraic expression equivalent to the given expression? ›

Combine any like terms on each side of the equation: ‍ -terms with ‍ -terms and constants with constants. Arrange the terms in the same order, usually ‍ -term before constants. If all of the terms in the two expressions are identical, then the two expressions are equivalent.

What is a pronumeral example? ›

Pronumerals are the letters used in algebra and they stand for numbers. Repeated occurrences of the same pronumeral in an expression represent the same value. Example. In the equation. x + x + x = 12.

Why do we use pronumerals? ›

Variables or pronumerals are introduced to students as a way of representing numbers using letters. Students may have previously seen symbols such as squares used to represent unknown numbers but will now be introduced to common pronumerals such as x, y and t.

How to solve substitution method? ›

Solve a system of equations by substitution
  1. Solve one of the equations for either variable.
  2. Substitute the expression from Step 1 into the other equation.
  3. Solve the resulting equation.
  4. Substitute the solution in Step 3 into one of the original equations to find the other variable.
  5. Write the solution as an ordered pair.

How do you substitute values for variables? ›

Substituting for a Variable

If you know the value that the variable is equal to, you can substitute that value in for the variable in the expression! Let's look at an example. Since we know that x=5, we can directly substitute or replace the x in the expression x+3 with a 5 to solve for y!

How to solve algebraic expressions step by step? ›

How to Solve an Algebra Problem
  1. Step 1: Write Down the Problem. ...
  2. Step 2: PEMDAS. ...
  3. Step 3: Solve the Parenthesis. ...
  4. Step 4: Handle the Exponents/ Square Roots. ...
  5. Step 5: Multiply. ...
  6. Step 6: Divide. ...
  7. Step 7: Add/ Subtract (aka, Combine Like Terms) ...
  8. Step 8: Find X by Division.

What is the simplified form of the algebraic expression? ›

An algebraic expression is in simplest form if it has no like terms and no parentheses. To combine like terms that have variables, use the Distributive Property to add or subtract the coefficients. The numerical factor of a term that contains a variable is a coefficient.

How do you simplify expressions in 7th grade? ›

An expression like 2x+3y+4x+7y can be simplified by combining like terms. Also, the terms 2x and 3y are unlike terms; we cannot simplify any expression with unlike terms because the variables are different. The term 2x= x+x or 2*x.. To simplify a linear expression, add or subtract like terms.

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