Solving Systems of Equations By Addition (Elimination)
After studying this lesson, you will be able to:
- Solve systems of equations by addition (elimination).
To Solve a System of Equations by Addition
(Elimination):
1. Add the equations together to eliminate one variable.
(Write the equations one over the other and add them together...add
straight down.)
2. Solve for the remaining variable.
3. Substitute the solution into one of the original equations
and solve for the other variable.
Example 1
Solve x + y = 5, x - y = 1
1 st : Line up the equations and add straight down to
eliminate a variable:
x + y = 5
x - y = 1
2x = 6
Solve 2x = 6
x = 3
2 nd : Now we substitute x = 3 into the first equation.
x + y = 5
3 + y = 5
y = 2
3 rd : Now we substitute the solution ( y = 2 ) into the other
equation:
x - y = 1
x - 2 = 1
x = 3
The solution is (3, 2)
Example 2
Solve 2x + 3y = 12, -2x + 9y = 12
1 st : Line up the equations and add straight down to
eliminate a variable:
2x + 3y = 12
-2x + 9y = 12
12y = 24 Solve this equation
y = 2
2 nd : Now we substitute y = 2 into the first equation.
2x + 3 (2) = 12
2x + 6 = 12
2x = 6
x = 3
3 rd : Now we substitute the solution ( x = 3 ) into the other
equation:
-2 (3) + 9y = 12
-6 + 9y =12
9y = 18
y = 2
The solution is (3, 2)
Example 3
Solve 3x + 4y = 19, 3x + 6y = 33
1 st : Line up the equations and add straight down to
eliminate a variable:
3x + 4y = 19
3x + 6y = 33
6x + 10 y = 52
Notice that a variable was not eliminated. In order to
eliminate a variable we need one of the 3x terms to be a
negative. We can do that by multiplying either equation by -1.
Let's multiply the first equation by -1 to get -3x - 4y = -19.
Now, let's add the equations together.
-3x - 4y = -19
3x + 6y = 33
2y = 14
y = 7
2 nd : Now we substitute y = 7 into the first equation.
-3x - 4 (7) = -19
-3x - 28 = -19
-3x = 9
x = -3
The solution is (-3, 7)
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