Elimination Method (Systems of Equations)
See It in Action
Two examples of the idea done right—and one common look-alike that is not right.
EXAMPLE
EXAMPLE
NON-EXAMPLE
Definition
The elimination method (also known as the addition method) is a technique used to solve a system of linear equations by eliminating one of the variables through addition or subtraction. The goal is to combine the equations in such a way that one variable cancels out, allowing you to solve for the other variable.
Steps for the Elimination Method:
Align the equations: Write the system of equations in standard form, where both equations are written as Ax + By = C.
Make the coefficients of one variable the same: If necessary, multiply one or both of the equations by a constant so that the coefficients of one of the variables are the same (or opposites).
Add or subtract the equations: Add or subtract the equations to eliminate one of the variables.
Solve for the remaining variable: Once one variable is eliminated, solve for the other variable.
Substitute back: After finding the value of one variable, substitute it back into one of the original equations to find the value of the other variable.
Example:
Consider the system of equations:
2x + 3y = 12
4x - 3y = 6
Step 1: Align the equations
2x + 3y = 12
4x - 3y = 6
Step 2: Make the coefficients of y opposites
Notice that the coefficients of y are already opposites (3 and -3), so we can proceed to eliminate y directly.
Step 3: Add the equations
Now, add the two equations together:
(2x + 3y) + (4x - 3y) = 12 + 6
This simplifies to:
6x = 18
Step 4: Solve for x
x = 18/6 = 3
Step 5: Substitute x = 3 back into one of the original equations
Substitute x = 3 into the first equation 2x + 3y = 12:
2(3) + 3y = 12
6 + 3y = 12
3y = 12 - 6 = 6
y = 6/3 = 2
Final Solution:
The solution to the system of equations is x = 3 and y = 2, or the point of intersection is (3, 2).
Why it's Called "Elimination":
The name "elimination" comes from the fact that you eliminate one variable by adding or subtracting the equations. Once one variable is eliminated, you can easily solve for the other variable.