5.2 Solving Systems of Equations by the Substitution Method

Solving Using Substitution

1. Solve either eqn. for x or y (may be done already).

2. Substitute the expression for the variable obtained in step 1 into the other eqn. and solve it.

3. Substitute the value for the variable from step 2 into any eqn. in the system that contains both variables and solve for the other variable.

4. Check soln. in BOTH eqns., if necessary.

Ex. Solve by the substitution method: x + y = 11y = 2x – 1

1. y = 2x – 12. x + y = 11

x + (2x – 1) = 11 x + 2x – 1 = 11 3x – 1 = 11 3x – 1 + 1 = 11 + 1

3x = 12 3x = 12 3 3 x = 4

3. y = 2x – 1y = 2(4) – 1 sub 4 for xy = 8 – 1y = 7

Soln: {(4, 7)}

4. Check:x + y = 11 y =

2x – 14 + 7 = 11 7 =

2(4) – 1 11 = 11 7 = 8 – 1

7 = 7

Ex. Solve by the substitution method: 2x – 2y = 2 x – 5y = -7

1. x – 5y = -7x – 5y + 5y = -7 + 5y x = 5y – 7

2. 2x – 2y = 22(5y – 7) – 2y = 2

10y – 14 – 2y = 2 8y – 14 = 2

8y + 14 – 14 = 2 + 14 8y = 16

8y = 16 8 8 y = 2

3. x = 5y – 7 x = 5(2) – 7 sub 2 for yx = 10 – 7 x = 3

Soln: {(3, 2)}

4. Check: 2x – 2y = 2 x – 5y = -72(3) – 2(2) = 2 3 – 5(2)= -

7 6 – 4 = 2 3 – 10 = -7

2 = 2 -7 = -7

Ex. Solve by the substitution method: x + 2y = -3

First: eliminate fractions by mult. by LCD

1. x – 3y = 2x – 3y + 3y = 2 + 3y x = 3y + 2

2. x + 2y = -33y + 2 + 2y = -3 5y + 2 = -3 5y + 2 – 2 = -3 – 2

5y = -5 5y = -5

5 5 y = -1

3. x = 3y + 2x = 3(-1) + 2 sub -1 for yx = -3 + 2 x = -1

Soln: {(-1, -1)}

3

1

26

yx

23

3

16

26

66

3

1

26

yx

yx

yx

Ex. Solve by the substitution method: -4x + 4y = -8-x + y = -2

1. -x + y = -2-x + y + x = -2 + x y = x – 2

2. -4x + 4y = -8

-4x + 4(x – 2) = -8 -4x + 4x – 8 = -8 -8 = -8

No variables remain and a TRUE stmt.

Lines coincide, so there are infinitely many solns.

Soln: {(x, y)| -x + y = -2} or {(x, y)| -4x + 4y = -8}

Ex. Solve by the substitution method: 6x + 2y = 7y = 2 – 3x

1. y = 2 – 3x

2. 6x + 2y = 7 6x + 2(2 – 3x) = 7 6x + 4 – 6x = 7 4 = 7 No variables remain and a

FALSE stmt.

lines are parallel, so there is NO SOLN. (empty set)

no soln. or ø

Groups

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