Study Guide: Section 8-2

Solving Systems by Substitution

One method of solving systems of equations algebraically is by substitution.

Example: Solve x + 3y = 7 and 2x − 4y = −6

Step 1: Solve the first equation for x in terms of y:

x + 3y = 7  &implies;  x = 7 − 3y

Step 2: Substitute (7 − 3y) for x in the second equation and solve for y:

2(7 − 3y) − 4y = −6
14 − 6y − 4y = −6
−10y = −20
y = 2

Step 3: Substitute 2 for y in either original equation to find the value for x:

x + 3(2) = 7
x + 6 = 7
x = 1

The solution to this system of equations is the ordered pair (1, 2).

Practice Problems

Directions: Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

1.

x = 3
2y + x = 3

Ans: (3, 0)

2.

y = 2
2x − 4y = 1

Ans: (9/2, 2)

3.

y = 3x − 7
3x − y = 7

Ans: infinitely many

4.

y = −x + 3
2y + 2x = 7

Ans: no solution

5.

x + y = 16
2y = −2x + 2

Ans: no solution

6.

x = 2y
0.25x + 0.5y = 10

Ans: (20, 10)

Word Problems

Directions: Set up a system of equations and solve by substitution.

7. How much of a 10% saline solution should be mixed with a 20% saline solution to obtain 1,000 milliliters of a 12% saline solution?

Ans: 800 mL of 10% and 200 mL of 20%

8. The tens digit of a two-digit number is 3 greater than the units digit. Eight times the sum of the digits is 1 less than the number. Find the number.

Ans: 41