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Lesson 6-3 Elimination Method
Elimination Method – using addition or subtraction to eliminate one variable when the two
equations are combined
Steps
1. Write the system so like terms with coefficients the same or opposite coefficients are
aligned.
2. Add or subtract the equations, eliminating one variable. Then solve the equation.
3. Substitute the value from step 2 into one of the equations and solve for the other variable.
4. Write the two variables as an ordered pair.
Example 1
Elimination Using Addition
Use elimination to solve the system of equations.
-4x + 3y = 11
4x + y = 9
 Step 1 -4x and 4x have opposite coefficients.
Step 2
Add the equations.
-4x + 3y = 11
(+) 4x + y = 9
4y = 20
The x variable is eliminated.
4y
20
=
4
4
Divide each side by 4.
y=5
Simplify.
Step 3
Substitute 5 for y in either equation to find the value of x.
4x + y = 9
Second equation
4x + 5 = 9
Replace y with 5.
4x + 5 – 5 = 9 – 5
Subtract 5 from each side.
4x = 4
Simplify.
4x 4
Divide each side by 4.
4 =4
x=1
Simplify.
Step 4
The solution is (1, 5).
Example 2 Write and Solve a System of Equations
Six times one number added to twice another number is –14. Negative six times the first
number minus the other number is 16. Find the numbers.
Six times one
number
added
to
twice another
number
is
-14.
6x
+
2y
=
-14
Negative six times
the first number
minus
the other
number
is
16.
–6x
–
y
=
16
Steps 1 and 2
Write the equations vertically and add.
6x + 2y = -14
(+) -6x – y = 16
y= 2
The variable x is eliminated.
Step 3
Substitute 2 for y in either equation to find the value of x.
-6x – y = 16
-6x – 2 = 16
-6x – 2 + 2 = 16 + 2
-6x = 18
–6x 18
–6 = –6
x = –3
Step 4 The numbers are –3 and 2.
Second equation
Replace y with 2.
Add 2 to each side.
Simplify.
CHECK
First equation
6x + 2y = -14
?
6(–3) + 2(2)  -14
-14 = -14 
–6x – y = 16
?
–6(–3) – 2  16
16 = 16 
Divide each side by –6.
Simplify.
Substitute –3 for x and 2 for y.
Simplify.
Second equation
Substitute –3 for x and 2 for y.
Simplify.
Example 3 Elimination Using Subtraction
Use elimination to solve the system of equations.
7a + 3b = 3
2a + 3b = 18
Step 1
Since 3b and 3b are the same, you can eliminate b by using
subtraction. Write the system so like terms are aligned.
Step 2
Subtract the equations.
7a + 3b = 3
(–) 2a + 3b = 18
5a
= -15
5a –15
5 = 5
a = -3
7a + 3b = 3
(–) 2a + 3b = 18
The variable b is eliminated.
Divide each side by 5.
Simplify.
Step 3
Substitute –3 for a in either equation to find the value of b.
2a + 3b = 18
Second equation
2(-3) + 3b = 18
a = -3
-6 + 3b = 18
Simplify.
3b = 24
Add 6 to each side and simplify.
3b 24
Divide each side by 3.
3 = 3
b=8
Simplify.
Step 4
The solution is (-3, 8).
Real-World Example 4 Write and Solve Systems by Elimination
EXERCISE Selena rode her bike 7 more miles yesterday than she did today. She rode
her bike a total of 23 miles yesterday and today. How many miles did she ride her bike
each day?
Understand
You know how many more miles she rode yesterday than today and the total
number of miles.
Plan
Let y = the miles that Selena rode yesterday.
Let t = the miles that Selena rode today.
yesterday’s
miles
y
yesterday’s
miles
y
Solve
plus
today’s
miles
equals
23.
+
t
=
23
minus
today’s
miles
equals
7.
–
t
=
7
Subtract the equations to eliminate one of the variables. Then solve for the
other variable.
y + t = 23
(+) y – t = 7
2y
= 30
2y 30
2 = 2
y = 15
Write the equations vertically and add.
The variable t is eliminated.
Divide each side by 2.
Simplify.
Now substitute 15 for y in either equation to find the value of t.
y + t = 23
First equation
15 + t = 23
Substitute 15 for y.
t=8
Subtract 15 from each side.
Selena rode her bike 15 miles yesterday and 7 miles today.
Check
To check these solutions, substitute both values into the other equation to see if
the equation holds true. If y = 15 and t = 8, then 15 – 8 = 7.