Elimination by Addition Method (Combination Method)
Solving a system of equations by substitution can be messy when none of the coefficients is 1. Fortunately, there is another way. We can always add the two equations to eliminate one of the variables. Sometimes, though, we need to multiply one or both equations by a number to make it work.
Example
Solve the systems of equations. Put your solutions in the form of a point, (x, y) .

Add the equations by adding like terms. Because we will be adding −3 y to 3 y , the y term will cancel, leaving one equation with only one variable.
We can put x = 2 into either A or B to find y . We will put x = 2 into A.
2 x − 3 y = 16
2(2) − 3 y = 16
−3 y = 12
y = −4
The solution is (2, −4).
Multiplying to Cancel Variables
Sometimes we need to multiply one or both equations by some number or numbers so that one of the variables cancels. Multiplying both sides of any equation by a nonzero number never changes the solution.
Examples

Because the coefficients on y are the same, we only need to make one of them negative. Multiply either A or B by −1, then add.
The solution is (2, −3).
Several options will work. We could multiply A by −2 so that we could add −4 x (in −2A) to 4 x in B. We could multiply A by 2 and multiply B by 7 so that we could add 14 y (in 2A) to −14 y (in 7B). We could also divide B by −2 so that we could add 2 x (in A) to −2 x (in − B). We will add −2A + B.
The solution is (4, −1).
Both equations in each of the following systems will need to be changed to eliminate one of the variables.
Examples

There are many options. Some are 3A − 8B, −3A + 8B, and 2A + 5B. We will compute 2A + 5B.
The solution is (1, 2).

First, we will eliminate the fractions. The LCD for A is 72, and the LCD for B is 30.
48 x − 18 y = 25 72A
15 x + 12 y = −1 30B
Now we will multiply the first equation by 2 and the second by 3.
Practice problems for this concept can be found at: Systems of Equations and Inequalities Practice Test.
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 PreCalculus: Systems of Equations and Inequalities
 PreCalculus: The Slope and Equation of a Line
 PreCalculus: Introduction to Functions
 PreCalculus: Functions and their Graphs
 PreCalculus: Combinations of Functions and Inverse Functions
 PreCalculus: Quadratic Functions
 PreCalculus: Polynomial Functions
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