Name _____________________________________________________ Date ____________ Color____________
Algebra I
Ms. Hahl
Solving Systems of Equations
Solving Systems with Linear Combinations (“Elimination”):
Sometimes solving a system of equations using substitution can be very difficult. For these problems we solve
using Linear Combinations (or Elimination). With elimination you solve by eliminating one of the variables. This
is accomplished by adding the 2 equations together. Before you can add the equations together, you need one
of the two variables to have two things:
1) Same Coefficient
2) Different Signs (one positive and one negative)
When you add terms with the same coefficient and different signs, the term drops out. You then solve for the
variable that is left. After you have solved for one variable, you plug the value into one of the original equations
and solve for the 2nd variable (just like Substitution). Then, you check the solution in both original equations.
The only difference between Substitution and Elimination is how you solve for the 1st variable. After that they
are the same.
Examples:
A) Sometimes it works out that the 2 equations already have a variable with the same coefficient and different
signs. You can then just add the equations:
First Variable
Second Variable
Check
SOLUTION:
B) Sometimes (usually) the equations do not have same coefficient and different signs, so we have a little bit of
manipulating to do.
First Variable
Second Variable
Check
SOLUTION:
C. Sometimes we need to manipulate both equations. We can do this by “criss crossing the coefficients.”
This is different than Example B, because no coeffcient
goes into another evenly.
First Variable
You need the negative sign to change the 6x to negative
so the signs will be different.
**You can also use 5 and -6. You can also “criss cross” the y coefficients.
Second Variable
Check
SOLUTION:
Practice:
Extra Problems – Solve each system algebraically:
1) 5x - 2y = -9
2) -4x + 2y = -16
7x + 2y = -27
5x – 3y = 19
3) x = 2y -6
5y –3x = 11
5) 4x – 5 = y
7x + 5y = 83
6) 7x + 4y = -11
5x + 2y = - 13
7) 5x – 6y = -17
3x + 8y = -16
8) x = 6 + 2y
6x – 5y = 15
9) 6x + 5y = 23
11x + 4y = 34
10) y = 3x + 4
8x – 9y = 59
11) 12x – 7y = 48
4x + 3y = -6
12) 9x – 4y = -88
2x + 5y = 4
13) 24x – 6y = -66
12x – 3y = -33
14) 5x – 6y = 42
15x – 18y = 54
15) 7x + 6y = -12
5x + 2y = -20
16) 13x – 3y = 78
4x + 6y = -66
17) 2y – 5 = x
4x – 11y = -38
18) 3x – 7y = -10
5x + 12y = -64
19) 6x – 17y = -104
4x – 7y = -39
20) 9x – 5y = -43
3x + 11y = 87
4) 5x –6y=- 74
7x + 5y = 17