Number and Algebra 53

Example 26 Solving simultaneous equations using substitution

Solve these pairs of simultaneous equations using the method of substitution.

a y = -3x + 2 and y = 7x – 8 b 2x – 3y = -8 and y = x + 3

SOLUTION EXPLANATION

a y = -3x + 2 (1) y = 7x – 8 (2) Substitute equation (2) into equation (1). 7x – 8 = -3x + 2

10x = 10 x = 1

Substitute x = 1 into equation (1). y = -3(1) + 2

= -1 Solution is (1, -1).

Write down and label each equation.

Alternatively substitute equation (1) into equation (2).Solve the equation for x.

Substitute the solution for x into one of the equations. Mentally check the solution using equation (2): -1 = 7(1) − 8.

b 2x − 3y = -8 (1) y = x + 3 (2) Substitute equation (2) into equation (1). 2x − 3(x + 3) = -8 2x – 3x – 9 = -8 -x − 9 = -8 -x = 1 x = -1 Substitute x = -1 into equation (2). y = -1 + 3

= 2 Solution is (-1, 2).

Label your equations.

Substitute equation (2) into equation (1) since equation (2) has a variable as the subject.Expand and simplify then solve the equation for x.

Substitute the solution for x into one of the equations to find y.

State the solution and check by substituting the solution into the other equation, (1).

1 By substituting the given point into both equations decide whether it is the solution to these simultaneous equations.a x + y = 5 and x − y = -1, point (2, 3)b 3x − y = 2 and x + 2y = 10, point (2, 4)c 3x + y = -1 and x − y = 0, point (-1, 2)d 5x − y = 7 and 2x + 3y = 9, point (0, 3)e 2y = x + 2 and x − y = 4, point (-2, -6)f x = y + 7 and 3y − x = 20, point (-3, 10)g 2(x + y) = -20 and 3x − 2y = -20, point (-8, -2)h 7(x − y) = 6 and y = 2x − 9, point (1, 6)

Exercise 1I

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ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.

© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press

Chapter 1 Linear relations54

2 Determine the point of intersection of each pair of equations by plotting accurate graphs.a y = 3 b y = -2 c x = 2 y = 2x − 4 y = 2x − 3 y = 4d x = -1 e y = x + 3 f y = 2x − 6 y = 0 2x + 3y = 14 3x − y = 7g y = 3x + 9 h y = x i y = -2x + 3 3x − y = 3 y = x + 4 y = 3x + 4

3 This graph represents the rental cost $C after k kilometres of a new car from two car rental firms called Paul’s Motor Mart and Joe’s Car Rental.

0 200 k

C

60

120

400 600 800 1000 1200

80100

140160180200220240260280 Joe’s Paul’s

a i Determine the initial rental cost from each company.ii Find the cost per kilometre when renting from each company.iii Find the linear equations for the total rental cost from each company.iv Determine the number of kilometres for which the cost is the same from both rental firms.

b If you had to travel 300 km, which company would you choose?c If you had $260 to spend on travel, which firm would give you the most kilometres?

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4 Solve the following pairs of simultaneous equations by using the method of substitution.a y = 4x + 2 and y = x + 8 b y = -2x – 3 and y = -x – 4c x = y – 6 and x = -2y + 3 d x = -7y − 1 and x = -y + 11

e y = 4 – x and y = x – 2 f y = 5 − 2x and y x= −32

2

g y = 5x – 1 and yx= −11 3

2 h y = 8x – 5 and y

x= +5 136

Example 26a

ISBN 978-0-521-17866-2 Photocopying is restricted under law and this material must not be transferred to another party.

© David Greenwood, Sara Woolley, Jenny Goodman, Jennifer Vaughan, GT Installations, Georgia Sotiriou, Voula Sotiriou 2011 Cambridge University Press