1. Solving Linear Systems by Linear Combinations
7.3
Solving Linear Systems by Linear Combinations
Objective 1 – Use linear combinations to solve a system of linear equations.
Steps to solve systems with linear combination.
1. Arrange equations with like terms in columns.
2. Combine like terms to get a variable to cancel.
3. Solve for one of the variables
4. Use the solution in step 3 to find the other variable.

2. Example
3x – 2y = 1
2x + 2y = 4
Combine like terms in
columns. The y’s cancel
5x = 5
Divide by 5
x = 1
Now substitute 1 for x in
one of the original equations
to solve for y.
2x + 2y = 4
2(1) +2y = 4
2 + 2y = 4
2y = 2
y = 1
Solution is (1,1)

3. EXAMPLE x + 4y = 23 – x + y = 2 Combine like terms 5y = 25 y = 5
Use y = 5 to solve for x
x + 4y = 23
x + 4(5) = 23
x + 20 = 23
x = 3
Solution (3, 5)

4. EXAMPLE Using Multiplication x – y = –5 x + 2y = 4
Nothing Cancels! We must multiply one of
the equations by a value to get opposites
Multiply one equation by by -1
– x + y = 5
x + 2y = 4
–1(x – y = –5)
x + 2y = 4
3y = 9
y = 3
x – y = –5
x – 3 = – 5
Solution (-2, 3)
x = – 2