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CHAPTER

10 Solving Linear Equations

GET READY 524

Math Link 526

10.1 Warm Up 527

10.1 Modelling and Solving One-Step

Equations: ax = b, xa

= b 528

10.2 Warm Up 537

10.2 Modelling and Solving Two-Step Equations: ax + b = c 538

10.3 Warm Up 548

10.3 Modelling and Solving Two-Step

Equations: xa

+ b = c 549

10.4 Warm Up 557

10.4 Modelling and Solving Two-Step Equations: a(x + b) = c 558

Chapter Review 568

Practice Test 574

Wrap It Up! 578

Key Word Builder 579

Math Games 580

Challenge in Real Life 581

Answers 582

524 MHR ● Chapter 10: Solving Linear Equations

Name: _____________________________________________________ Date: ______________

A letter that represents an unknown number

What you do to 1 side, do to the other.

Rewrite as 4x = 24

Substituting Values into Equations To substitute values into an equation: Step 1: Replace the variable.

Step 2: Follow the order of operations: • brackets first • multiply and/or divide in order from left

to right • add and/or subtract in order from left to

right

Find the value of y when x = 5. y = 2(x – 4) + 3 Substitute. y = 2(5 – 4) + 3 Brackets first. y = 2(1) + 3 Multiply. y = 2 + 3 Add. y = 5

1. Find the value of y when x = 4.

a) y = 2(x – 1) + 6 y = 2(4 – 1) + 6

y = 2( ) + 6

y = + 6

y =

b) y = (3 + x – 7) × 4

y = (3 + – 7) × 4

y = ( – 7) × 4

y = × 4

y =

Modelling and Solving One-Step Equations To solve a word problem, change the words into symbols, letters, and numbers to make

an equation.

A number increased by 5 is 8. 4 times a number is 24. n + 5 = 8 4 × x = 24

To solve an equation, use the opposite operation to get x alone on 1 side of the equation.

opposite operation • an operation that undoes another operation • – and + are opposite operations, × and ÷ are opposite operations

Examples: n + 5 = 8 n + 5 – 5 = 8 – 5 n + 5 – 5 = 8 – 5 n = 3

4x = 24 44x 24

4=

x = 6

Get Ready ● MHR 525

Name: _____________________________________________________ Date: ______________

Get x alone on 1 side of the

equation.

2. Write each sentence as an equation.

a) Seven more than a number is twelve.

n + =

b) Three less than a number is eleven.

x − =

c) Four times a number is twenty-eight.

d) When a number is divided by nine, the result is nine.

Solving Two-Step Equations

To solve a 2-step equation, find the value of x. 2x + 3 = 7

2x + 3 – 3 = 7 – 3 Subtract to undo addition. 2x + 3 – 3 = 7 – 3 2x = 4 Divide to undo multiplication.

22x 4

2=

x = 2 3. For each equation, circle the first operation you should do to both sides.

a) 2n + 4 = 18 Add 4 or subtract 4.

b) 3x + 5 = 17 Add 5 or subtract 5.

c) 8y – 70 = 94 Add 70 or subtract 70.

d) 27 = 7q + 6 Add 6 or subtract 6.

4. Solve the equations.

a) 5j + 9 = 29

5j + 9 – = 29 –

5j =

5 j=

j =

b) 2t – 2 = 14


Name: _____________________________________________________ Date: ______________

Modelling Equations You can use equations to encrypt a password.

equation • 2 expressions joined with an equal sign • examples: x – 5 = 8 or 2y + 3 = 15

Jim’s password is weather. He gives each letter in the alphabet a number value.

1 = a 2 = b 3 = c 4 = d 5 = e 6 = f 7 = g 8 = h 9 = i 10 = j 11 = k 12 = l 13 = m 14 = n 15 = o 16 = p 17 = q 18 = r 19 = s 20 = t 21 = u 22 = v 23 = w 24 = x 25 = y 26 = z

w e a t h e r The number sequence for the password weather is: 23 5 1 20 8 5 18

Jim uses the equation y = 3x + 2 to encrypt the password. This gives each letter in the password a different number value. Find the new values of the letters. w = 23 e = 5 a = 1 y = 3(23) + 2 y = 3( ) + 2 y = 69 + 2 y = + 2 y = y = t = h = r =

Jim’s code for weather is 71 .

change a word into a code

10.1 Warm Up ● MHR 527

Name: _____________________________________________________ Date: ______________

× =+ ++

× =- +-

× =- -+

× =+ --

represents +1represents x

represents –1

x10.1 Warm Up 1. Write an equation for each diagram.

a) _______________________________

b) _______________________________

c) _______________________________

d) _______________________________

2. Show each equation using algebra tiles.

a) 3x + 5 = –1 b) 2x + 1 = 5

3. Multiply or divide.

a) 6 × (–5) =

b) 3 × 8 =

c) (–4) × (–7) =

d) (–24) ÷ 6 =

e) 15 ÷ 3 = f) (–12) ÷ (–4) = 4. Fill in the boxes.

a) + 4 = 7 b) + 8 = 10

c) – 5 = 6 d) – 1 = 4

e) × 2 = 12 f) × 3 = 15

g) ÷ 4 = 10 h) ÷ 6 = 3

xxx

= x =

xxx

=x =


Name: _____________________________________________________ Date: ______________

What number times 3 equals

–12?

represents a positive variable.

represents a negative variable.

represents +1. represents –1.

10.1 Modelling and Solving One-Step Equations: ax = b, xa

= b

linear equation • an equation whose points on a graph lie along a straight line

• examples: y = 4x, d = 2c , 5w + 1 = t

numerical coefficient • a number that multiplies the variable • example: 4k + 3 numerical coefficient

variable • a letter used to represent an unknown number • example: 2y + 3 variable constant • a number that is not connected to a variable • example: 2x + 5 = –3 constant

Working Example 1: Solve an Equation

Solve each equation. a) 3x = –12

Solution

Method 1: Solve by Inspection

3x = –12

3 × = –12

3 × 4− = –12

So, x =

Method 2: Solve Using Models and Diagrams Use algebra tiles.

�

x

x

x 3 black tiles = 3x 12 white tiles = –12 3 black tiles = 12 white tiles

How many white tiles = 1 black tile?

So, x = .


= b ● MHR 529

Name: _____________________________________________________ Date: ______________

r ÷ –2

What number divided by –2 equals –7?

b) 72

r =−−

Solution Method 1: Solve by Inspection

72r= −

−

72

= −−

147

2= −

−

So, x = . Check:

Left Side Right Side

2

r−

= 2−

=

–7

Method 2: Solve Using Models and Diagrams 1 whole circle = –r Divide the circle into 2 parts. Shade half the circle.

__�r2

Add white tiles to show –7.

� �1 �1 �1 �1 �1 �1 �1__�r2

Half the circle shows 2r− , so multiply the

amount of shaded circle by 2 to represent –r. Then, multiply the number of white squares by 2 so that the equation is balanced.

�

�1�1

�1�1

�1�1

�1�1

�1�1

�1�1

�1�1

�r

white squares represent –14.

So, –r = .

To make r positive, multiply both sides of the equation by –1.

–1(–r) = –1(–14)

So, r = .


Name: _____________________________________________________ Date: ______________

Solve each equation. Use inspection, models, or diagrams. Check your work. Remember,

a) 4x = –8

b) 52y=

Check:

Left Side Right Side 4x

= 4 ×

=

–8

Check: Left Side Right Side

2y

= 2

=

5

× =+ ++

× =- +-

× =- -+

× =+ --


= b ● MHR 531

Name: _____________________________________________________ Date: ______________


–84

12d = 12 (–7) = ______

Working Example 2: Divide to Apply the Opposite Operation

Simone uses a spring to do an experiment. The equation F = 12d shows how much force the spring has. F = force in newtons, N d = distance the spring stretches, in cm Simone pushes the spring together with a force of 84 N. What distance is the spring compressed? Solution F = 12d Substitute –84 for F. –84 = 12d The opposite of × 12 is 12.

84 12d−= C 84 ÷ 12 = ?

= d The spring is compressed cm.

Solve by applying the opposite operation. Check your answer. a) –5b = –45

5 45b− −=

b =

b) 6f = –12


–5b

= –5 ×

=

–45


The spring was pushed in, so d is negative.

pressed together


Name: _____________________________________________________ Date: ______________

Working Example 3: Multiply to Apply the Opposite Operation

In January, the average afternoon temperature in Edmonton is 13

the average

afternoon temperature in Yellowknife. The average afternoon temperature in Edmonton is –8 °C. What is the average afternoon temperature in Yellowknife? Solution Let t represent the average afternoon temperature in Yellowknife. 13

the average afternoon temperature in Yellowknife = 13

t× or 3t

The equation to model the problem is 3t = –8.

3t = –8 The opposite of ÷ 3 is 3.

3t 3× 8 3= − ×

t = The average afternoon temperature in Yellowknife is °C.

Solve by applying the opposite operation. Check your answer.

35

d=

−

_______ 3 _______5

_______

d

d

× = ×−

=


5

d−

= 5−

=

3


3t

= 243−

=

–8


= b ● MHR 533

Name: _____________________________________________________ Date: ______________

represents +1 represents –1

1. Five times a number is –15.

a) Write an equation for this sentence.

b) Draw a picture to show how to find the unknown number.

2. Raj is solving the equation 9n = –4.

n__9

= –4

n__9

X (–9) = -4 X (–9)

n = 36

Raj’s solution is wrong. Explain where he made his mistake. ____________________________________________________________________________

3. Write the equation modelled by each diagram.

a)

t =

b)

c)

3m =

d)

�

t

t

t

�

�n

�n�__m3

�1 �1

�__�w2

�1 �1 �1 �1


Name: _____________________________________________________ Date: ______________

4. Solve by inspection.

a) –2j = 12

–2 × = 12

j =

b) 5n = –25

5 × = –25

n =

5. Solve each equation using the opposite operation. Check your answers.

a) 4s = –12

4 12

______________

s

s

−=

=

Check:


b) –36 = –3j Check:


c) 3t = –8

× 3t = –8 ×

t = Check:


d) 9 = 10x

−

× 9 = 10x

− ×

= x Check:



= b ● MHR 535

Name: _____________________________________________________ Date: ______________

6. Nakasuk’s snowmobile can travel for 13 km on 1 L of gas.

a) How far can he travel on 2 L of gas?

b) How far can he travel on 3 L of gas?

c) How far can he travel on 10 L of gas?

d) Nakasuk visits his aunt who lives 312 km away.

How many litres of gas will he need? Use the equation 13x = 312 to help you.

7. Lucy is making 2 pairs of mittens. She has 72 cm of trim to sew around the cuffs. How much trim does she have for each mitten?

a) How many mittens in total is she making? b) Let t represent the amount of trim for 1 mitten.

Equation to find the amount of trim for 1 mitten: t =

c) Solve the equation to find the amount of trim for 1 mitten. Sentence: _________________________________________________________________


Name: _____________________________________________________ Date: ______________

Have you ever dropped a ball of Silly Putty® on a hard surface? It bounces! The greater the height from which it is dropped, the higher it bounces. With a partner, perform an experiment with Silly Putty®.

a) Tape a metre stick vertically (↕) to a wall or desk. The 0-cm end should touch the floor. One person sits on the floor looking at the ruler. The other person stands next to the ruler, holding the putty. b) Choose 4 different heights from which to drop the putty. Record the heights in the table. c) Drop the ball of putty from the 4 different heights you chose in part b). Each time, the person on the floor watches to see the height the putty bounces. The person on the floor records the height of the bounce in the table. Round your answer to the nearest centimetre. d) Complete the chart to find k. Round your answers to 2 decimal places.

Trial Height of Drop,

h Height of First Bounce,

b k = b

h

Example 20 cm 3 cm k = 320

= 0.15

1

2

3

4

e) What do you notice about the numbers you recorded in the last column? _________________________________________________________________________ f) Write an equation that shows the results of your experiment. Substitute your value for k into the equation. b = h


Name: _____________________________________________________ Date: ______________

_____ ÷ 3 = 7

3(2) = 3 × 2 = 6

10.2 Warm Up 1. Write the equation for each diagram.

a)

b)

2. Evaluate.

a) b + 6, when b = 5 b) s – 3, when s = 12

b + 6

= + 6

=

3. Find the missing number by inspection. a) 5 × = 15 b) 4 × = 24

c) ÷ 2 = 8 d) 73

=

4. Multiply.

a) 10(–4) = b) –6(–1) =

5. Calculate. Use the number line to help you.

�15 �13�14 �10�11�12 �9 �8 �5�6�7 �4 �1�2�3 0 �1 �2 �3 �4 �5 �6 �7 �8 �9 �12�10�11 �15�13�14

a) (–5) + (+8) = b) (+11) + (–4) =

c) (–6) + (–7) = d) (–3) + (+12) =

e) (+8) – (–6) f) (–5) – (–9)

= (+8) + (+6) ← Add the opposite → = (–5) + ( )

= =

g) (–9) – (+3) h) (+2) – (+12)

x =

xxx

=


Name: _____________________________________________________ Date: ______________

the number that multiplies the variable

10.2 Modelling and Solving Two-Step Equations: ax + b = c Working Example 1: Modelling With a Balance Scale

Solve the equation 23 = 4k + 3. Solution You can use a balance scale to model this equation. The numerical coefficient is , so draw 4 black tiles.

k k

k k

Isolate the variable by removing grey tiles from the right. To keep things balanced, remove grey tiles from the left.

k k

k k

There are k blocks on the right side of the scale. There are grey blocks on the left side of the scale.

To balance the scale, each k block must have a mass of grey blocks. So, k = .


23 4k + 3 = 4(5) + 3

= 20 + 3 =

10.2 Modelling and Solving Two-Step Equations: ax + b = c ● MHR 539

Name: _____________________________________________________ Date: ______________

Use the balance scale to solve each equation. Show your steps, then check your answer. a) 6n + 6 = 18

So, n = . Check:

Left Side Right Side 6n + 6 = 6( ) + 6 = + 6 =

18

b) 13 = 9 + 2p

Check:

Left Side Right Side 13

9 + 2p = 9 + 2( ) = 9 + =


Name: _____________________________________________________ Date: ______________

unknown value = number of hours an elephant sleeps

the number that multiplies the variable

The negative 1-tile and positive 1-tile on the left side equal 0.

Working Example 2: Model With Algebra Tiles

A cow sleeps 7 h a day. This is 1 h less than twice the amount an elephant sleeps in a day. How long does an elephant sleep?

a) Write an equation for this situation. Solution Let e represent the number of hours an elephant sleeps. “Twice what an elephant sleeps” = 2e “1 h less than twice what an elephant sleeps” = 2e – A cow sleeps 7 h, so the equation is 2e – 1 = b) Solve the equation using algebra tiles. Solution The numerical coefficient is , so draw black tiles.

�

e

e To isolate the black tiles on 1 side of the equal sign, add grey tile to both sides.

�

e

e

black tile is equal to grey tiles.

�

�e

e e = 4 So, an elephant sleeps h a day.


Name: _____________________________________________________ Date: ______________

Solve each equation using algebra tiles. a) 2x + 4 = –6 Fill in the missing tiles so your model represents the equation.

=

Isolate the black tiles. Split up the tiles so you can see that 1 black tile is equal to white tiles. x = b) 3r – 2 = 13 r =


Name: _____________________________________________________ Date: ______________

Working Example 3: Apply the Opposite Operations

Solve 4w + 3 = 19 by applying the opposite operation. Solution To isolate a variable, follow the reverse order of operations. 4w + 3 = 19 4w + 3 – 3 = 19 – 3 Subtract 3 from both sides of the equation.

4w = Divide both sides by 4.

44x

4=

w = Check using algebra tiles: Draw algebra tiles to represent the equation. Isolate the black tiles.

Subtract grey 1-tiles from both sides. The 4 black tiles must have the same value as the 16 grey 1-tiles.

Each black tile must have a value of grey 1-tiles.

w =

Solve by applying the opposite operation.

a) –3x – 7 = 5

–3x – 7 + = 5 +

–3x =

3x−=

x =

b) 4 + 2g = –12

4 – + 2g = –12 – ( )

2g = –12 + ( )

2g =

=

g =

+ and – × and ÷

=wwww

=wwww

====

wwww


Name: _____________________________________________________ Date: ______________

Isolate means to get the variable alone.

1. a) Draw algebra tiles to model 3x – 5 = 16.

b) To isolate x, add tiles to both sides of the equation. Give 1 reason why you need to add this number of tiles. Hint: Use zero pairs. _________________________________________________________________________ _________________________________________________________________________ c) To solve for x, divide both sides of the equation by . Give 1 reason why you need to divide by this number. _________________________________________________________________________ _________________________________________________________________________

2. Solve each equation modelled by the algebra tiles.

a) x =

b) t =

�

x

x

x

x

=t

t


Name: _____________________________________________________ Date: ______________

3. Solve the equation modelled by each balance scale. Check your solution.

a) 3x + 8 = 11

x

x

x

x =

b) 13 = 4h + 5

h =

Remove 8 grey tiles from each side.


Name: _____________________________________________________ Date: ______________

4. Complete the table.

Equation First Operation to Solve Second Operation to Solve 4r – 2 = 14 Add ________ to each side. Divide both sides by ________.

–22 = –10 + 2n

53 = –9k – 1

3 – 3x = –9 5. Solve each equation and check your answer.

a) 6r + 6 = 18 6r + 6 – = 18 – 6r =

6r=

r = Check:

Left Side Right Side 6r + 6 = 6( ) + 6 = + 6 =

18

b) 4m – 2 = 14



Name: _____________________________________________________ Date: ______________

w = width

m = money in Jennifer’s account

6. You buy lunch at Sandwich Express.

A sandwich costs $4. Each extra topping costs $2. You have $10. Use the equation 2e + 4 = 10 to find how many extra toppings you can get if you spend all of your money.

Sentence: ____________________________________________________________________ 7. Jennifer is saving money to buy a new bike. She doubled the money in her bank account, and then she took out $50. She has $300 left in her account.

a) Write an equation to find the amount b) Solve the equation.

in her account at the beginning.

_____________ 300m − =

Jennifer had

in her bank account.

8. A classroom’s length is 3 m less than 2 times its width. The classroom has a length of 9 m.

a) Write an equation to find the width of the classroom.

Equation: w – =

b) Solve the equation to find the width of the classroom. Sentence: _________________________________________________________________

MENUYour choice of extras,only $2 each:salad, fries, milk, juice, jumbo cookie,frozen yogurt.

l = 9

w

10.2 Math Link ● MHR 547

Name: _____________________________________________________ Date: ______________

As an object falls, it picks up more and more speed. A falling object increases its speed by about 10 m for every second it falls.

a) Find the speed at which an object hits the ground for different starting speeds and different amounts of time that the object falls.

Complete the table.

Starting Speed (m/s)

Amount of Time the Object Falls

(s)

Speed at Which It Hits the Ground = Starting Speed + (10 × Time Falling)

(m/s)

0 5 0 + (10 × 5) = 50

5 4 5 + (10 × 4) =

10 1 + (10 × ) =

0 12 + (10 × ) =

15 3

b) A stone comes loose from the side of a canyon. It falls with a starting speed of 5 m/s. It hits the water at a speed of 45 m/s. Find the amount of time the stone falls before it hits the water. Speed at which it hits the water = starting speed + (10 × time falling) = + 10t Sentence: ________________________________________________________________

Imagine falling 10 m in 1 s!


Name: _____________________________________________________ Date: ______________

10.3 Warm Up 1. Use the opposite operation to solve.

a) p – 5 = 20 p – 5 + 5 = 20 + p =

b) x + 9 = 14

c) 3x = 15

3 15x=

x =

d) 7 = 5d

3. Draw a model for each equation. Do not solve. a) x – 7 = 5

b) 5g + 3 = –1

4. Multiply or divide.

a) 7(2) =

b) 5(–4) =

c) 162

= d) 155− =

5. Subtract.

a) 9 – (–3) = + Add the opposite. =

b) –13 – 6 = (–13) – (+6) Rewrite. = + Add the opposite. =

10.3 Modelling and Solving Two-Step Equations: xa

+ b = c ● MHR 549

Name: _____________________________________________________ Date: ______________


+ b = c

Working Example 1: Model Equations

The elevation of Qamani’tuaq, Nunavut, is 1 m less than 12

the elevation of Prince Rupert, B.C.

If the elevation of Qamani’tuaq is 18 m, what is the elevation of Prince Rupert?

a) Write an equation to find the elevation of Prince Rupert. Solution Let p represent the elevation of Prince Rupert.

“ 12

the elevation of Prince Rupert” = 12

p

“The elevation of Qamani’tuaq is 1 m less than 12

” = 12

p – 1

The elevation of Qamani’tuaq is 18 m, so 12

p – 1 = .

b) Draw a diagram to solve the equation. Check your answer. Solution 12

p – 1 = 18

The constants in this equation are and 18. To isolate the variable, add positive grey square to both sides.

The 12

circle is equal to grey squares.

12

p =

× 12

p = 19 × Multiply both sides by .

p = 38 The elevation of Prince Rupert is m.

Elevation is the height above sea level.

a number that is not connected to a variable

BritishColumbia

Prince Rupertelevation = �

Nunavut

Qamani’tuaqelevation = 18 m

__p2

�

�1

�1�1

�1

�1

�1

�1

�1

�1

�1

�1

�1

�1

�1

�1

�1

�1

�1

�1 �1

�1

__p2

� �1

�1

�1

�1

�1

�1

�1

�1

�1

�1

�1

�1

�1

�1

�1

�1

�1 �1

�1

__p2

� �1

�1

�1

�1

�1

�1

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�1

�1

�1

�1

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�1

�1

�1

�1

�1


Name: _____________________________________________________ Date: ______________

You need 4 equal parts to fill the circle.

Solve by modelling each equation.

a) 4x – 5 = –7

�

�1�1 �1

�1 �1 �1 �1�1 �1 �1 �1 �1

To isolate the variable, add grey tiles to both sides.

__x4

=

14

of the circle is equal to white squares.

14

x =

× 14

x = ×

x =

b) 3x + 1 = –4 c) 3 +

2k = –5

+ is a zero pair and represents 1 + (–1) = 0


+ b = c ● MHR 551

Name: _____________________________________________________ Date: ______________

Working Example 2: Apply the Reverse Order of Operations

Kristian Huselius played for the Calgary Flames during the 2006–2007 NHL season.

He had 41 more than 12

the number of shots on goal as Jarome Iginla.

Huselius had 173 shots on goal. How many shots on goal did Iginla have?

a) Write an equation to find the number of shots on goal

Jarome Iginla had. Solution Let j represent the number of shots on goal Jarome Iginla had.

“Huselius had 41 more than 12

the number of shots on goal” = 2j + .

Since Huselius had 173 shots, 2j + 41 = .

b) Solve the equation to find Jarome Iginla’s number of shots on goal. Solution

2j + 41 = 173

2j + 41 – = 173 – Subtract from both sides.

2j =

22j

× = 2 × Multiply both sides by .

j = Jarome Iginla had shots on goal during the 2006–2007 season. Check:


2j + 41

= 2

+ 41

= + 41

=

173


Name: _____________________________________________________ Date: ______________

Solve by applying the reverse order of operations.

a) 12

x− – 6 = 4

12

x− – 6 + = 4 +

12

x−× = × 12

–x = 120 The numerical coefficient of x is .

1 1201x−=

−

x =

b) –4 = 3 + 7k

Subtract 3 from both sides. Multiply both sides by 7.


+ b = c ● MHR 553

Name: _____________________________________________________ Date: ______________

1. Describe how to isolate the variable when solving 5n – 12 = 6.

____________________________________________________________________________

____________________________________________________________________________

2. Manjit thinks that the first step in solving the equation 4x−

+ 7 = 9 is to multiply both sides of

the equation by –4. He writes:

x – ––

×( 4)+7 =9×( 4)4

Is he correct? Circle YES or NO.

Give 1 reason for your answer. ___________________________________________________

____________________________________________________________________________

3. Write the equation for each diagram and name the constants.

Diagram Equation Constants

a)

b)

c)

d)

A constant is a number that is not connected to a variable.

�

�1

�1 �1

�1 �1�1

�1__x3

�

�1

�1

�1

�1

�1

�1

�1

�1

�1__�b2

�

�1

�1

�1

�1

�1

�1

�1�1

�1

�1

�1__z5

�

__d7

�1�1

�1

�1

�1

�1

�1

�1

�1�1 �1


Name: _____________________________________________________ Date: ______________

4. Draw a model for 5 35g− = . Then, solve and check your answer.

Model: Check:


2g – 5 3

Solution: g =

5. Solve each equation using the reverse order of operations. Check your answers.

a) 2 + 3m = 18

2 – + 3m = 18 –

3m =

3m

× = ×

m = Check:


b) 8

c−

– 8 = –12

Check:



+ b = c ● MHR 555

Name: _____________________________________________________ Date: ______________

Show $2 less than 12

of

the adult cost.

6. People 18 years old or younger need a certain number of hours of sleep each day.

The equation s = 12 – 4a tells you how many hours of sleep they need.

s = amount of sleep needed, in hours a = age of the person, in years

a) If Brian needs 10 h of sleep, how old is he?

s = 12 – 4a

b) Natasha is 13 years old. She gets 8 h of sleep a night. Is this enough sleep?

s = 12 – 4a

7. The cost of a concert ticket for a student is $2 less than 12

of the cost for an adult.

a) Write an expression for the cost of a concert ticket for a student. a = the cost for an adult

Cost of student ticket = a

b) If the cost of a student concert ticket is $5, how much does the adult ticket cost? Equation: Sentence: _________________________________________________________________


Name: _____________________________________________________ Date: ______________

The atmosphere is the air that surrounds Earth. It is made of many different layers. The troposphere is the lowest layer of the atmosphere, where humans live and where weather occurs. Look at the graph.

a) What pattern do you see in the graph? ________________________________ ________________________________ ________________________________

b) The equation that models air temperature change in the troposphere is t = 15 – 154

h ,

where t is the temperature, in °C, and h is the height, in metres. At what height in the troposphere is the temperature 0 °C?

15154

0 15154

0 _______ 15 _______ Subtract 15 from both sides.154

_______154

_______ _______ _______ Multiply both sides by 154.154

_______

1 Divide both sides by 1.

_______

ht

h

h

h

h

h

h

h

= −

= −

− = − −

−=

−× = ×

= −

−= −

=

Sentence: ________________________________________________________________

mesosphere

ozone layer stratosphere

troposphere 14 000 m

tropopause

Earth

t

0

Air Temperature in the Troposphere

2000 4000 6000 8000 10 000h

−60

−50

−40

−30

−20

−10

10

20


Name: _____________________________________________________ Date: ______________

10.4 Warm Up 1. Fill in the blanks.

a) The opposite of adding 4 is b) The opposite of dividing by 6 is c) The opposite of multiplying by –4 is d) The opposite of subtracting 7 is

2. Solve each equation.

a)

b)

3. Solve.

a) 3(4 – 2) = 3( ) =

b) 2(3 × 4) = 2( ) =

4. Solve.

a) (–5) – (–4) = (–5) + ( ) =

b) 4 – (–6)

c) (–12) + (–3) = d) 7 + (–8) =

e) (–6) ÷ (–3) = f) (–20) ÷ 4 =

g) 7 × (–3) = h) (–8) × (–2) =

=�


Name: _____________________________________________________ Date: ______________

10.4 Modelling and Solving Two-Step Equations: a(x + b) = c Working Example 1: Model With Algebra Tiles

A flower garden is in the shape of a rectangle. The length of the garden is 2 m longer than the length of the shed. The width of the garden is 4 m. The area of the garden is 20 m2. What is the length of the shed?

a) Write an equation. Solution Let s represent the length of the shed. “The length of the garden is 2 m longer than the length of the shed” = s + Area = 20 m2, so the equation is 20 = 4 × ( ) or 4(s + 2) = 20. b) Solve the equation. Solution

Model the equation 4(s + 2) = 20 using algebra tiles. There are 4 groups of s + 2. That means there are 4 black tiles and 8 positive 1-tiles on the left side of the equation. To isolate the variable, add white 1-tiles to each side of the equation. Now, there are 4 black tiles on the left and 12 positive 1-tiles on the right. Each black tile equals 3 positive 1-tiles, so s = . The length of the shed is m.

Area of a rectangle = length × width

equals zero. Remove all the zeros.

A = 20 m2

s + 2

4

=

s

s

s

s

=

ssss

=

s

s

s

s

2 m

A = 20 m2

4 m

s

10.4 Modelling and Solving Two-Step Equations: a(x + b) = c ● MHR 559

Name: _____________________________________________________ Date: ______________

Solve by modelling the equation. a) 2(g + 4) = –8 Draw 2 groups of g + 4 on the left side and 8 white 1-tiles on the right. Add 8 white 1-tiles to both sides of the diagram. Remove the zero pairs. Draw the result.

Group the tiles into 2 equal groups by circling the equal groups above. Each black tile equals negative 1-tiles. g = b) 3(r – 2) = 3 Draw 3 groups of r – 2 on the left side and 3 positive 1-tiles on the right. Add 6 positive 1-tiles to both sides of the diagram. Isolate the variable then draw the result. Group the tiles into 3 equal groups by circling the equal groups above. Each variable tile equals positive 1-tiles. r =


Name: _____________________________________________________ Date: ______________

Working Example 2: Solve Equations

Kia is making a square quilt with a 4-cm border around it. She wants the quilt to have a perimeter of 600 cm. Find the lengths of the sides of Kia’s quilt before she adds the border.

a) Write an equation. Solution Let s represent the length of the side before the border is added. A 4-cm border is added to each end of the side length: s + There are 4 sides to the quilt: perimeter = 4(s + 8) The perimeter is 600 cm, so the equation is 4(s + 8) = . b) Solve the equation to find the side length of the quilt. Solution distributive property

• a(b + c) = a × b + a × c • when you multiply each term inside the brackets by the term outside the brackets • example: 2(x + 3) = (2 × x )+ (2 × 3)

= 2x + 6

Method 1: Divide First 4(s + 8) = 600

4 ( 8) 6004

s +=

s + 8 = 150 s + 8 – = 150 – s = The side length of the quilt before the border is added is cm.

Method 2: Use the Distributive Property 4(s + 8) = 600 4s + 32 = 600 4s + 32 – = 600 –

44 4s=

s = ________ The quilt dimensions before adding the border are 142 cm × 142 cm.

P of a square = 4s

s + 8

4 cm

s

4 cm


Name: _____________________________________________________ Date: ______________

c) Solve –4(x – 5) = 24. Method 1: Divide First –4(x – 5) = 24

4 ( 5) 244x− −

=−

Divide by – 4.

x – 5 = (– )

x – 5 + 5 = (– ) + 5 Add to both sides.

x = Method 2: Use the Distributive Property First –4(x – 5) = 24 Multiply x and –5 by –4. (–4x) + (–4)( –5) = 24 –4x + = 24 –4x +20 – 20 = 24 – 20 Subtract from both sides. –4x =

4 44x−=

− Divide both sides by –4.

x = Check:


–4 (x – 5) = –4 (–1 – 5) = –4 (–6) = 24

24

÷ =+ ++

÷ =- +-

÷ =- -+

÷ =+ --


Name: _____________________________________________________ Date: ______________

a) Solve –2(x – 3) = 12 by dividing first.

2( 3) 12x− −= Divide both sides by –2.

(x – 3) = ( ) x – 3 + = ( ) + Add 3 to both sides. x = b) Solve –20 = 5(3 + p) using the distributive property. –20 = 5(3 + p) Multiply (3 + p) by 5. –20 = (5 × ) + ( 5 × ) Multiply. –20 = + –20 – ( ) = 15 – + Subtract 15 from both sides. –20 + ( ) = 5p Add the opposite. = 5p Divide both sides by 5.

5p=

p = c) Solve 5(x – 3) = –15.


Name: _____________________________________________________ Date: ______________

1. Julia and Chris are solving the equation –6(x + 2) = –18.

a) Julia: Is she correct? Circle YES or NO. Give 1 reason for your answer. ________________________________________________________________________ ________________________________________________________________________

b) Chris: Is he correct? Circle YES or NO. Give 1 reason for your answer. ________________________________________________________________________ ________________________________________________________________________

2. Model each equation with algebra tiles.

a) 3(t – 2) = 12

b) 6( j – 1) = –6

c) 2(3 + p) = 8 d) 14 = 7(n – 2)

First, I subtract 2 from both sides. Then, I divide both sides by –6.

I start by dividing –6(x + 2) by –6. Then, I subtract 2 from both sides.


Name: _____________________________________________________ Date: ______________

3. Solve the equation modelled by each diagram. Check your answers. a) Check:


b) Check:


4. Solve each equation by dividing first. Check your answers. a) 6(r + 6) = –18

( )6 6 18r + −=

r + 6 = r + 6 – = 3 – r = Check:


b) 4(m – 3) = 12

m = Check:


=

x

x

x

x

=x

x


Name: _____________________________________________________ Date: ______________

5. Solve each equation using the distributive property. a) 21 = 3(k + 3) 21 = 3(k + 3) Multiply k + 3 by 3.

21 = +

21 – = + – Subtract 9 from both sides.

=

k

= Divide both sides by 3.

k = b) 40 = –4(n – 3) 40 = –4(n – 3) Multiply n and –3 by –4.

40 = ( ) + 12

40 – = (– ) + 12 – Subtract 12 from both sides.

= ( )

28 4n−= Divide both sides by –4.

n = c) 8(x – 3) = 32 d) 3(1 + g) = 27


Name: _____________________________________________________ Date: ______________

6. An old fence around Gisel’s tree is shaped like an equilateral triangle.

Gisel wants to build a new fence. She wants to make each side 7 cm longer. She wants the perimeter to be 183 cm. a) Write an equation for this problem. f = length of fence before adding 7 cm

The length, f, with 7 cm added = Since all 3 sides are equal, the equation is 3( f + 7) = b) Solve the equation to find the length of each side of the old fence. The old fence measures along each of its sides.

7. The formula E = –125(t – 122) shows the amount of energy a hiker needs each day on a hike. E is the amount of food energy, in kilojoules (kJ), and t is the outside temperature,

in degrees Celsius. a) If the outside temperature is –20 °C, how much food energy will the hiker need each day? Sentence: ________________________________________________________________ b) If a hiker uses 16 000 kJ of food energy, what is the outside temperature? Sentence: ________________________________________________________________

Does –20 replace E or t?

the sides are the same length

Does 16 000 replace E or t?

perimeter

10.4 Math Link ● MHR 567

Name: _____________________________________________________ Date: ______________

In some jobs, people work the night shift. In other jobs, people work alone or with dangerous materials. Sometimes you get paid more money when you have to work under certain conditions. a) A worker makes r dollars per hour. His boss asks him to work the night shift and offers him a premium of $2 more per hour. What expression represents his new pay? b) If he works 6 h, the expression 6(r + 2) represents his pay.

What is the expression if he works for 8 h? c) The worker makes $184 for working an 8-h night shift. Write an equation for this problem.

d) How much does he make per hour when he works the night shift? The worker makes per hour on the night shift. e) Research 1 job that pays an hourly or monthly wage plus a premium. Briefly describe the job and explain why a premium is paid for the job. ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________

This increase in your pay is called a premium.

First solve for r. Then add $2.


Name: _____________________________________________________ Date: ______________

10 Chapter Review Key Words For #1 to #7, choose the word from the list that goes in each blank. variables distributive property equations constants numerical coefficients opposite operations linear equations 1. Letters that represent unknown numbers are called . 2. are made up of 2 expressions that are equal to each other. 3. Multiplication and division are of

each other. 4. Numbers that are attached to a variable by multiplication are called . 5. 5(b + 3) = 5 × b + 5 × 3 is an example of how you use the . 6. In the equation 2x – 7 = 5, both –7 and 5 are . 7. Equations that, when graphed, result in points that lie along a straight line are called

.

10.1 Modelling and Solving One-Step Equations: ax = b, ax

= b, pages 528–536

8. Solve by inspection.

a) 6r = –18

6 × = –18

r = _______________

b) –5 = p3

–5 = ÷ 3

p = ____________

Chapter Review ● MHR 569

Name: _____________________________________________________ Date: ______________

9. Solve the equation modelled by each diagram. Check your answers.

a) Check: Left Side Right Side

b) Check:


10. Solve each equation using the opposite operation.

a) –5t = 15

5 15t−=

t =

b) 2

a−

= –7

2

a−

× = –7 ×

a =

10.2 Modelling and Solving Two-Step Equations: ax + b = c, pages 538–547 11. Write and solve the equation modelled by the diagram. Check your answer.

Check:


=x

x

x

c c

c

−1 −1=__n4


Name: _____________________________________________________ Date: ______________

12. Solve each equation using opposite operations.

a) –3t + 8 = 20

–3t + 8 – = 20 –

–3t =

3t−=

t =

b) 5j – 2 = –12

13. Zoë has a collection of CDs and DVDs. The number of CDs is 3 fewer than 4 times the number of DVDs. Zoë has 25 CDs.

a) Write an equation for this situation. Let d represent the DVDs. number of CDs = 3 fewer than 4 times the DVDs:

b) Solve the equation.

Zoë has DVDs.


+ b = c, pages 549–556

14. Solve the equation modelled by the diagram.

Diagram after isolating the variable: j =

Equation:

= −1 −1−1 −1 −1

__j4


Name: _____________________________________________________ Date: ______________

15. Solve. Check your answers.

a) 3d – 13 = –8

3d – 13 + = –8 +

3d × = ×

d = Check:


b) 17 = –4 + 2x−

Check:


16. An airplane ticket is on sale for $350. The sale price is one third of the regular price, plus $100 in taxes.

a) Write an equation to represent this situation. Let r = the regular price.

b) What is the regular price of the airplane ticket? Solve the equation.

Sentence: ___________________________________________________________________

Sale price = 13

of the regular price + $100


Name: _____________________________________________________ Date: ______________

10.4 Modelling and Solving Two-Step Equations: a(x + b) = c, pages 558–567 17. Solve the equation modelled by each diagram. Check your answers.

a) Check:


b) Check:


18. Solve. Check your answers.

a) 6(q – 13) = 24

6( 13) 24q −=

q = __________ Check:


b) 2(g + 4) = 14

Check:


r

= r

r

=w

w


Name: _____________________________________________________ Date: ______________

19. Solve using the distributive property.

a) –5(w – 4) = 10

–5(w – 4) = 10 + = 10 + – = 10 – = 10 + ( ) =

=

w =

b) 4(m – 3) = 12

20. Each side of a square is decreased by 3 cm. The perimeter of the new square is 48 cm. What is the length of each side of the original square?

a) Write an equation. The length of the each side of the original square is s.

length of the each side after decreasing it by 3 cm =

perimeter of the new square = Since all 4 sides are equal, the equation is 4( ) = .

b) Solve the equation to find the length of each side of the square.

Sentence: _________________________________________________________________


Name: _____________________________________________________ Date: ______________

10 Practice Test For #1 to #4, circle the best answer.

1. What is the solution to 3x = –12?

A x = 36 B x = 4 C x = –4 D x = –36

2. What is the solution to 5n + 6 = –4?

A n = –10 B n = 5 C n = –2 D n = 2

3. Which of these equations has the solution p = –6?

A 3p – 4 = –2 B

3p + 4 = –2

C 3p−

– 4 = –2 D 3p−

+ 4 = –2

4. Wanda solved the equation 4(x – 3) = 2 like this:

4(x – 3) = 2 Step 1 4x – 12 = 8 Step 2 4x = 20 Step 3 x = 5 In which step did Wanda make her first mistake? A Step 1 B Step 2 C Step 3 D No mistake was made.

Complete the statements in #5 and #6. 5. The opposite operation of division is . 6. The solution to 4(y + 5) = 24 is y = .

Practice Test ● MHR 575

Name: _____________________________________________________ Date: ______________

Short Answer 7. Dillon used algebra tiles to model a problem.

=

xx

a) What equation is modelled?

b) Using the algebra tiles, what is the first step that Dillon should take to solve the equation? ________________________________________________________________________

8. a) Draw algebra tiles to model the equation 3x + 5 = –7.

b) Draw the tiles you need to make zero pairs.

Explain why you need those tiles. ________________________________________________________________________

c) To solve for x, divide both sides of the equation by the numerical coefficient . Explain why you would divide by this number. ________________________________________________________________________

d) Solve the equation.


Name: _____________________________________________________ Date: ______________

9. Solve each equation. Check your answers.

a) 4x = 48 Check:


b) 5t−

= –8

Check:


c) 2k – 6 = 12 Check:


d) 12 = 4(x – 2)

Check:


Practice Test ● MHR 577

Name: _____________________________________________________ Date: ______________

Use the distributive property or divide first.

10. Beth would like to put a 2-m-wide grass border around a square garden. The perimeter of the outside of the border is 44 m. a) Write an equation for this situation. Let s be the length of the side of the garden. Perimeter of large square = 4 × length

= 4 (s + )

b) Find the length of each side of the garden. Solve your equation.

________________________________________________________________________ 11. The formula for the perimeter of a rectangle is P = 2(l + w).

P is the perimeter, l is the length, and w is the width. Find the length of the rectangle if P = 14 cm and w = 3 cm.

Formula → P = 2(l + w)

Equation → = 2(l + ) Solve →


Sentence: __________________________________________________________________

w = 3 cm

l

2 m

s +

s


Name: _____________________________________________________ Date: ______________

How are linear equations used in everyday life? For each linear equation in the table: • Identify what each variable, numerical coefficient, and constant represents. • Solve each equation. Show your steps on a separate sheet of paper. • Name a job or a real-life situation where each equation could be used.

Equation Type Equation

Identify the Variables (V),

Coefficients (CO), and Constants (C)

Solve the Equation

Job or Real-Life Situation

Example: ax = b 15t = 45 t : V 15: CO 45: C

15t = 45 15 × 3 = 45

t = 3

Finding the time it took to finish a race.

ax = b 3d = 27

xa

= b 58p=

ax + b = c 2 7 19c+ =

xa

+ b = c 11 73v− =

a(x + b) = c 6( 5) 36w+ =

Key Word Builder ● MHR 579

Name: _____________________________________________________ Date: ______________

Use the clues to identify key words from Chapter 10. Then, write them in the crossword puzzle. Across 6. In 5d + 4 = 39, 5 is the . 7. When solving an equation such as 6a – 4 = 26, use the to isolate the variable.

8. In the equation 7m = 6, m is the .

Down 1. In the equation 5w + 1 = t, 1 is a . 2. 5(s + 2) = 5s + 10 uses the property. 3. Two mathematical expressions joined by an equal sign is an . 4. When graphed, a equation results in points along a straight line. 5. When solving 6a = 72, you need to the variable.

1

2 3

4

5

7

8

6


Name: _____________________________________________________ Date: ______________

Math Games

Rascally Riddles What do mathematicians use to shampoo carpets? To answer the riddle, solve each of the equations. Replace each box with a letter. The letter you put in the box will be the variable that has the same solution as the number above the box. The first one has been done for you.

4 2 4 4 6 4 3

3 2 6 7 3 1 2 1 3

2c = –8 2 82 2c −=

c = –4

5o – 1 = 9

3e = 2

4r−

+ 3 = 2

4i – 3 = 1 3(s – 5) = –6

3l + 4 = 2 4t + 7 = –5

2(n + 1) = 0 7

u−

= 1

c c

Challenge in Real Life ● MHR 581

Name: _____________________________________________________ Date: ______________

Challenge in Real Life

Earth’s Core

Earth is made up of layers. The deeper layers are hotter because the temperature inside Earth increases the deeper you go.

• grid paper • coloured pencil

The table shows how the temperature increases with depth.

Layer Depth (km) Temperature (°C) Crust –90 870 Mantle –2921 3700 Outer core –5180 4300 Inner core –6401 7200

1. On a separate sheet of grid paper, graph the data from the table. 2. Use the formula T = d + 870 to find the approximate temperature

at any depth where d is the depth, in km, and T is the temperature in ºC. a) Calculate the approximate temperature

if the depth is 3400 km.

T = d + 870

T = + 870

T =

The temperature would be at a depth of 3400 km.

b) Check your solution by graphing it on your graph from #1. Does this point lie in a line with the other points?

Circle YES or NO.

c) If the temperature is approximately 9000 ºC, how deep are you beneath Earth’s surface?

T = d + 870

= d + 870

The depth is if the temperature is approximately 9000 ºC.

d) Using a different coloured pencil, check your solution by graphing it on your graph from #1. Does this point lie in a line with the other points? Circle YES or NO.

InnerCore

Outer Core

Crust

Centre of Earth

870 °C

3700 °C

4300 °C

7200 °C

Mantle


Answers

Get Ready, pages 524–525

1. a) y = 12 b) y = 0

2. a) n + 7 = 12 b) x – 3 = 11 c) 4x = 28 d) 99x=

3. a) subtract b) subtract c) add d) subtract 4. a) j = 4 b) t = 8 Math Link

Jim’s code is 71, 17, 5, 62, 26, 17, 56. 10.1 Warm Up, page 527

1. a) 3x + 6 = 9 b) x – 8 = 5 c) x = 5 d) 3x = 6 2. a) b) 3. a) –30 b) 24 c) 28 d) –4 e) 5 f) 3 4. a) 3 b) 2 c) 11 d) 5 e) 6 f) 5 g) 40 h) 18


= b,

pages 528–536

Working Example 1: Show You Know

a) x = –2 b) y = 10 Working Example 2: Show You Know

a) b = 9 b) f = –2 Working Example 3: Show You Know

d = –15 Communicate the Ideas

1. a) 5x = –15 b)

2. Answers may vary. Example: Raj multiplied both sides by –9, but he should have used +9.

Practise

3. a) 3t = –6 b) 2w− = –4 c)

3m− = –2 d) –2n – 10

4. a) j = –6 b) n = –5 5. a) s = –3 b) j = 12 c) t = –24 d) x = –90 Apply

6. a) 26 km b) 39 km c) 130 km d) 24 L 7. a) 4 b) 4t = 72 c) 18 cm Math Link

d) Answers may vary. Example:

Trial Height of Drop,

h Height of First Bounce,

b k = b

h

20 3.0 0.15 1 18 2.7 0.15 2 16 2.4 0.15 3 10 1.5 0.15 4 5 0.8 0.15

e) Answers may vary. Example: All the numbers in the last column are the same. f) b = 0.15h

10.2 Warm Up, page 537

1. a) x – 3 = 4 b) 3x – 2 = 4 2. a) 11 b) 9 3. a) 3 b) 6 c) 16 d) 21 4. a) –40 b) 6 5. a) +3 b) +7 c) –13 d) +9 e) +14 f) +4 g) –12 h) –10 10.2 Modelling and Solving Two-Step Equations: ax + b = c,

pages 538–547


a) n = 2 b) p = 2 Working Example 2: Show You Know

a) x = –5 b) r = 5 Working Example 3: Show You Know

a) x = –4 b) g = –8 Communicate the Ideas

1. a) b) 5 grey; You add 5 to both sides to get the x-tiles alone. c) 3; You

divide by 3 to get 1 x tile. Practise

2. a) x = 3 b) t = –7 3. a) x = 1 b) h = 2 4.

Equation First Operation to Solve Second Operation to

Solve 4r – 2 = 14 Add 2 to each side. Divide each side by 4. –22 = –10 + 2n Add 10 to each side. Divide each side by 2. 53 = –9k – 1 Add 1 to each side. Divide each side by –9. 3 – 3x = –9 Subtract 3 from each side. Divide each side by –3.

5. a) r = 2 b) m = 4 Apply

6. 3 toppings 7. a) 2m – 50 = 300 b) $175 8. a) 2w – 3 = 9 b) 6 m Math Link

a) b) 4 s Speed at Which It Hits the Ground(m/s)

50 45 20 120 45

+ =x

x

x

+ =x

x

=

=

x

x

xx

x

x

+

=

=x

x

x

x

x

r

r

r

−

=

=

r

− =x

x

x

Answers ● MHR 583


1. a) p = 25 b) x = 5 c) x = 5 d) d = 35 3. a) b) 4. a) 14 b) –20 c) 8 d) –3 5. a) +12 b) –19


+ b = c,

pages 549–556


a) x = –8 b) x = –15 c) k = –16 Working Example 2: Show You Know

a) x = –120 b) k = –49 Communicate the Ideas

1. Add 12 to both sides. Multiply both sides by 5. 2. NO. The first step is to subtract 7 from both sides. Practise

3. a) 3x – 2 = 5; –2, 5 b) 3 = –

2b – 6; –6, 3

c) 5z + 7 = 4; 7, 4 d) –3 =

7d –8; –8, –3

4. =

g2

+1

+1

+1

−1

−1

−1

−1 −1

g = 16

5. a) m = 48 b) c = 32 Apply

6. a) 8 years old b) No, she needs 8.75 hours of sleep a night.

7. a) c = 12

a – 2 b) $14

Math Link

a) For every 1000-m increase in height, the air temperature drops about 7 °C. b) 2310 m


1. a) subtracting 4 b) multiplying by 6 c) dividing by –4 d) adding 7 2. a) x = –2 b) x = 3 3. a) 6 b) 24 4. a) –1 b) +10 c) –15 d) –1 e) 2 f) –5 g) –21 h) +16 10.4 Modelling and Solving Two-Step Equations: a(x + b) = c, pages 558–567


a) g = –8

b) r = 3


a) x = –3 b) p = –7 c) x = 0 Communicate the Ideas

1. a) NO. She must divide by –6 first. b) NO. He should divide both sides by –6.

Practise

2. a) b)

c) d)

3. a) x = 6 b) x = 4 4. a) r = –9 b) m = 6 5. a) k = 4 b) n = –7 c) x = 7 d) g = 26 Apply

6. a) 3(f + 7) = 183 b) 54 cm 7. a) 17 750 kJ b) –6 °C Math Link

a) h(r + 2) b) 8(r + 2) c) 184 = 8(r + 2) d) $23 e) Answers will vary. Example: Nurses are paid a premium if they work shifts that are longer than 12 hours.

Chapter Review, pages 568–573

1. variables 2. equations 3. opposite operations 4. numerical coefficients 5. distributive property 6. constants 7. linear equations 8. a) r = –3 b) p = –15 9. a) x = –3 b) n = –8 10. a) t = –3 b) a = 14 11. 3c + 2 = 5; c = 1 12. a) t = –4 b) j = –2 13. a) 25 = 4d – 3 b) 7

14. –2 = 4j – 3; j = 4

15. a) d = 15 b) x = –42

16. a) 3r + 100 = 350 b) $750

17. a) r = –2 b) w = –5 18. a) q = 17 b) g = 3 19. a) w = 2 b) m = 6 20. a) 48 = 4(x – 3) b) 15 cm Practice Test, pages 574–577

1. D 2. C 3. C 4. A 5. multiplication 6. 1 7. a) 2(x – 4) = 6 or 2x – 8 = 6 b) Add 8 positive tiles to both sides or divide

both sides by 2 or multiply 2 by x and – 4.

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8. a) b) You need zero pairs to isolate the 3 x-tiles.

c) 3; You divide by 3 to isolate x. d) x = –4 9. a) x = 12 b) t = 40 c) k = 9 d) x = 5 10. a) 44 = 4(s + 4) b) 7 m 11. 4 cm Wrap It Up!, page 578

Equation Type Equation

Identify the Variables,

Coefficients, and Constants

Solve the Equation

Job or Real-Life Situation

ax = b 3d = 27 • d: V • 3: CO • 27: C

d = 9 Answers will vary. Example: Finding the force applied to a spring.

xa

= b 58p=

• p: V • 8: CO • 5: C

p = 40 Finding the temperature of one place compared to somewhere else.

ax + b = c 2c + 7 = 19 • c: V • 2: CO • 19: C

c = 6 Finding the wind speed of one place compared to another.

xa

+ b = c 11 73v− =

• v: V • 3: CO • 11 and 7: C

v = 57 Finding the elevation of one place compared to another.

a(x + b) = c 6(w + 5) = 36 • w: V • 6: CO • 5 and 36: C

w = 1 Finding the length of a square when you know the area and how the length compares to the width.

Key Word Builder, page 579

Across 6. numerical coefficient 7. opposite operation 8. variable Down 1. constant 2. distributive 3. equation 4. linear 5. isolate Math Games, page 580

e = 6, i = 1, l = –6, n = –1, o = 2, r = 4, s = 3, t = –3 u = –7 Challenge in Real Life, page 581

1.

2. a) 4270 ºC b) NO c) 8130 m d) YES

+ +

x

x

x

T

0

Depth vs. Temperature

Temperature (ºC)

7000

9000

6000

8000

2000

4000

1000

3000

5000

–2000–4000–6000Depth (km)

d

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