Substitution Method
Solve this system of equations:
3y - 2x = 11
y + 2x = 9
1. Solve one of the equations for either "x =" or "y =".
This example solves the second equation for "y ="
3y - 2x = 11
y = 9 - 2x
2. Replace the "y" value in the first equation by what "y" now equals. Grab the "y" value and plug it into the other equation.
3(9 - 2x) - 2x = 11
3. Solve this new equation for "x".
(27 - 6x) - 2x = 11
27 - 6x - 2x = 11
27 - 8x = 11
-8x = -16
x = 2
4. Place this new "x" value into either of the ORIGINAL equations in order to solve for "y". Pick the easier one to work with!
y = 9 - 2(2)
y = 9 - 4
y = 5
5. Check: substitute x = 2 and y = 5 into BOTH ORIGINAL equations. If these answers are correct, BOTH equations will be TRUE!
3y - 2x = 11
3(5) - 2(2) = 11
15 - 4 = 11
11 = 11
y + 2x = 9
5 + 2(2) = 9
5 + 4 = 9
9 = 9
Elimination:
x - 2y = 14
x + 3y = 9
a. First, be sure that the variables are "lined up" under one another. In this problem, they are already "lined up".
x - 2y = 14
x + 3y = 9
b. Decide which variable ("x" or "y") will be easier to eliminate. In order to eliminate a variable, the numbers in front of them (the coefficients) must be the same or negatives of one another. Looks like "x" is the easier variable to eliminate in this problem.
x - 2y = 14
x + 3y = 9
c. Now, subtract to eliminate the "x" variable.
(Remember: when you subtract signed numbers, you change the signs and follow the rules for adding signed numbers.)
x - 2y = 14
-x - 3y = - 9
- 5y = 5
d. Solve this simple equation.
-5y = 5
y = -1
e. Plug "y = -1" into either of the ORIGINAL equations to get the value for "x".
x - 2y = 14
x - 2(-1) = 14
x + 2 = 14
x = 12
f. Check: substitute x = 12 and y = -1 into BOTH ORIGINAL equations. If these answers are correct, BOTH equations will be TRUE!
x - 2y = 14
12 - 2(-1) = 14
12 + 2 = 14
14 = 14
x + 3y = 9
12 + 3(-1) = 9
12 - 3 = 9
9 = 9