Solving Systems of
Equations
The Elimination Method
Using Addition and Subtracting
• I can:
– solve a system of two linear equations in
two variables algebraically using the
Elimination Method
The Elimination Method
Using Addition
Objectives
• Solve systems of linear equations in
two variables by elimination.
• Compare and choose an appropriate
method for solving systems of linear
equations.
1
The Elimination Method
Using Addition
• So far, we have solved systems using
graphing and substitution. These notes
show how to solve the system
algebraically using ELIMINATION with
addition, subtraction and multiplying.
• Elimination is easiest when the equations
are in standard form.
• ax + by = c or 2x +3y = 12
The Elimination Method
You must align all like terms in the equations. Then
determine whether any like terms can be eliminated
because they have opposite coefficients.
Since –2y and 2y have opposite coefficients, the yterm is eliminated. The result is one equation that
has only one variable: 6x = –18.
2
Solving a system of equations by
elimination using addition and subtraction.
x+y=5
3x – y = 7
Step 1: Put the equations in
Standard Form.
They already are!
The y’s have the same
coefficient.
Step 2: Determine which
variable to eliminate.
Step 3: Add or subtract the
equations.
Add to eliminate y.
x+ y=5
(+) 3x – y = 7
4x
= 12
x=3
Solving a system of equations by
elimination using addition and subtraction.
x+y=5
3x – y = 7
Step 4: Plug back in to find
the other variable.
Step 5: Check your
solution.
x+y=5
(3) + y = 5
y=2
(3, 2)
(3) + (2) = 5
3(3) - (2) = 7
3
Elimination using Addition
Consider the system
x - 2y = 5
Lets add both equations
to each other
2x + 2y = 7
REMEMBER: We are trying to find the
Point of Intersection. (x, y)
Elimination using Addition
Consider the system
x - 2y = 5
+
2x + 2y = 7
Lets add both equations
to each other
NOTE: We use the Elimination Method, if we can immediately
cancel out two like terms.
4
Elimination using Addition
Consider the system
x - 2y = 5
+
2x + 2y = 7
= 12
3x
x=4 
Lets add both equations
to each other
ANS: (4, y)
NOTE: We use the Elimination Method, if we can immediately
cancel out two like terms.
Elimination using Addition
Consider the system
x - 2y = 5
2x + 2y = 7
4 - 2y = 5
- 2y = 1
y= 1 
2
Lets substitute x = 4 into this
equation.
Solve for y
ANS: (4, y)
NOTE: We use the Elimination Method, if we can immediately
cancel out two like terms.
5
Elimination using Addition
Consider the system
x - 2y = 5
Lets substitute x = 4 into this
equation.
2x + 2y = 7
4 - 2y = 5
- 2y = 1
y= 1 
2
Solve for y
1
ANS: (4, 2 )
NOTE: We use the Elimination Method, if we can immediately
cancel out two like terms.
Elimination using Addition
Consider the system
3x + y = 14
4x - y = 7
NOTE: We use the Elimination Method, if we can immediately
cancel out two like terms.
6
Elimination using Addition
Consider the system
3x + y = 14
+
4x - y = 7
7x
= 21
x=3 
ANS: (3, y)
Elimination using Addition
Consider the system
3x + y = 14
Substitute x = 3 into this equation
4x - y = 7
3(3) + y = 14
9 + y = 14
y=5
ANS: (3, 5 )
NOTE: We use the Elimination Method, if we can immediately
cancel out two like terms.
7
Elimination using Addition
Solve using the addition method.
x–y=7
x+y=3
x+y=3
5+y=3
2x + 0y = 10
y = -2
2x = 10
x=5
(5,-2)
Elimination using Addition
Solve using the addition method.
2x + 3y = 11
2x + 3y = 11
-2x + 9y = 1
2x + 3(1) = 11
0x + 12y = 12
2x + 3 = 11
12y = 12
2x = 8
y=1
x=4
(4,1)
8