7.2 Solving Systems of Linear Equations by Substitution

Leanne Thompson

7.2 Solving Systems of Linear Equations by Substitution

Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, in many cases, solving a system by graphing may be inconvenient or imprecise. If the graphs extend beyond the small grid, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph. In this section, we will solve systems of linear equations by substitution.

Solve a System of Equations by Substitution

Table 7.2.1
To solve a system of equations by substitution:
Step 1: Solve one of the equations for either variable. Step 2: Substitute the expression from Step 1 into the other equation. Step 3: Solve the resulting equation. Step 4: Substitute the solution in Step 3 into one of the original equations to find the other variable. Step 5: Write the solution as an ordered pair. Step 6: Check that the ordered pair is a solution to both original equations.

Example 1

Solve the system by substitution:
$2x + y = 7$
$x - 2y = 6$

Table 7.2.2
Steps	Solution
Solve one of the equations for either variable.	$2x + y = 7$ $y = 7 - 2x$
Substitute the expression from Step 1 into the other equation.	$x - 2y = 6$ $x - 2(7 - 2x) = 6$
Solve the resulting equation.	$x - 2(7 - 2x) = 6$ $x - 14 + 4x = 6$ $5x = 20$ $x = 4$
Substitute the solution in Step 3 into one of the original equations to find the other variable.	$2x + y = 7$ $2(4) + y = 7$ $8 + y = 7$ $y = -1$
Write the solution as an ordered pair.	(4, –1)
Check that the ordered pair is a solution to both original equations.	$2x + y = 7$ $2(4) + (-1) = 7$ $7 = 7$ $4 - 2(-1) = 6$ $6 = 6$

Practice 1

Solve the system by substitution.

a)

$x + y = -1$
$y = x + 5$

b)

$x + y = 6$
$y = 3x - 2$

c)

$-2x + y = -11$
$x + 3y = 9$

d)

$x + 3y = 10$
$4x + y = 18$

e)

$y = -2x + 5$
$y = \dfrac{1}{2}x$

f)

$4x + 2y = 4$
$6x - y = 8$

Homework

Solve the system by substitution.

a)

$y = -3x$
$y = 6x - 9$

b)

$y = x + 5$
$y = -2x - 4$

c)

$y = -2x - 9$
$y = 2x - 1$

d)

$y = 6x + 3$
$y = 6x - 1$

e)

$y = 6x + 4$
$y = -3x - 5$

f)

$y = -2x - 9$
$y = -5x - 21$
Solve the system by substitution.

a)

$y = 3x - 4$
$3x - 3y = -6$

b)

$y = 6x + 21$
$-x + 3y = 12$

c)

$y = -6$
$3x - 6y = 30$

d)

$2x - y = 1$
$y = -3x - 6$

e)

$3x + y = 5$
$2x + 4y = -10$

f)

$4x + y = 2$
$3x + 2y = -1$

g)

$-2x + 2y = 6$
$y = -3x + 1$

h)

$y = 4x - 1$
$8x - 2y = 6$

i)

$x - 2y = -2$
$3x + 2y = 34$

j)

$-6x + 6y = -12$
$8x - 3y = 16$

k)

$2x + y = -4$
$3x - 2y = -6$

l)

$-8x + 2y = -6$
$-2x + 3y = 11$

m)

$y = \dfrac{1}{2}x - 4$
$2x - 4y = 16$

n)

$-x - 4y = -14$
$-6x + 8y = 12$

o)

$5x - 3y = 2$
$y = \dfrac{5}{3}x - 4$
Solve the system of equations:
$x + y = 10$
$x - y = 6$
a) by graphing
b) by substitution
c) Which method do you prefer for this question?
Solve the system of equations:
$3x + y = 12$
$x = y - 8$
a) by graphing
b) by substitution
c) Which method do you prefer for this question?
Solve the system of equations using substitution:
$x + 2y = 7$
$x = 3y - 5$

a) x = $\dfrac{11}{5}$ , y = $\dfrac{12}{5}$
b) x = $\dfrac{3}{7}$ , y = $\dfrac{2}{3}$
c) x = 2, y = 3
d) x = 4, y = 1

Which of the following statements is true about the substitution method for solving systems of linear equations?

a) The substitution method is only useful when the system has one equation in terms of one variable.
b) The substitution method involves substituting one equation into another to eliminate one variable, making it easier to solve for the other variable.
c) The substitution method requires that both equations in the system be written in slope y-intercept form.
d) The substitution method requires finding a common denominator for the variables in the system.

Solve the system of equations using substitution:
$4x - 3y = 10$
$x = \frac{2y + 6}{3}$

a) x = –1, y = –3
b) x = 2, y = 3
c) x = 1, y = 4
d) x = –2, y = –6

Answers

1.

a) (1, –3)

b) (–3, 2)

c) (–2, –5)

d) No solution

e) (–1, –2)

f) (–4, –1)

2.

a) (3, 5)	b) (–3, 3)	c)	d) (–1, –3)	e)
f) (1, –2)	g) $\left(-\dfrac{1}{2},\dfrac{5}{2}\right)$	h) No solution	i) (8, 5)	j) (2, 0)
k) (–2, 0)	l) (2, 5)	m) Infinitely many solutions	n) (2, 3)	o) No solution

3. a) (8, 2)
b) (8, 2)
c) Answers will vary.

4. a) (1, 9)
b) (1, 9)
c) Answers will vary.

5. a

6. b

7. d

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resource(s):

Berg, T. (2020). Intermediate algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY-NC-SA 4.0

Marecek, L., Anthony-Smith, M., & Honeycutt, M. (2020). Elementary algebra 2e. OpenStax. https://collection.bccampus.ca/textbooks/elementary-algebra-2e-openstax-106/, licensed under CC BY 4.0

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