Elimination

1
Elimination Using Addition and Subtraction
2
Adding Opposites
Previously, we learned that adding two numbers
together which have the same absolute value but
are opposite in sign results in a value of zero.
This can be shown graphically using algebra tiles.
Each algebra tile has the same absolute value but
is opposite in sign. Each pair of tiles makes a
zero pair.
This same principle can be applied to variables.
Below is a zero pair for the variable, x.
3
Elimination
We can apply the idea of a zero pair to systems
of equations so that one of the variables can be
eliminated.
Look at the following system of equations. What
do you observe?
In each equation, there is a y. However, the
signs in the two equations are opposite. If
added together, the y and the -y would make a
zero pair.
4
Addition
Add the following two equations together by
adding like terms together.
Now pick one of the two equations and substitute
the value of x into that equation and solve
for y.
x y 6 4 y 6 -4 -4
Now solve for x.
y 2
2x 8
Write the solution as an ordered pair.
2 2
x 4
Solution (4,2)
5
Addition
Solution (4,2)
To check your work, substitute the values for the
variables into each equation and determine if it
is true.
?
?
The solution is correct.
6
Subtraction
Addition worked when the signs of one of the
variables were opposite. However, you may
encounter a system of equations in which the
signs of the variables are the same. In this
case, instead of adding, you will be subtracting
one equation from the other. Remember that
subtraction is really the same as adding the
opposite.
Lets take a look at the following system of
equations
The signs of the variables are all positive.
Therefore, in order to solve this system, we can
subtract the bottom equation from the top
equation.
7
Subtraction
Since we have the value for x. Pick one of the
equations and substitute the value 3 for x and
solve for y.
Subtract the bottom equation from the top
equation.
3x y 6 (3)(3) y 6 9 y
6 -9 -9
(-)
or
y -3
Write the solution as an ordered pair.
Solution (3,-3)
8
Subtraction
Solution (3,-3)
To check your work, substitute the values for the
variables into each equation and determine if it
is true.
?
?
The solution is correct.
9
No Solution
Solve the following system of equations by using
addition or subtraction.
When subtraction is used, there are no more
variables (x or y) remaining and the result is an
incorrect statement. We know that
This system of equations can be solved by using
subtraction.
Therefore, this system of equations represents
parallel lines and there is no solution.
(-)
10
Summary of Steps

1. Arrange the two equations so that the like terms
   are in vertical columns.
2. If the signs of the variables are opposite, add
   the two equations together to eliminate one of
   the variables.
3. If the signs of the variables are the same, then
   subtract one of the equations from the other
   equation.
4. Solve for the remaining variable.
5. Substitute the value of the variable into one of
   the equations and solve for the other variable.
6. Check the solution by substituting the values of
   the two variables into each equation.
7. Write the solution. If there is one solution,
   write it as an ordered pair.

11
You Try It
Solve each system of equations by using addition
or subtraction.

• The sum of two numbers is 85. The difference
  between
• the two numbers is 19. What are the two
  numbers?

12
Problem 1
Add the two equations together.
Solve for y by substituting the value for x
into one of the equations.
()
Solve for x.
3x 21
3
3
x 7
Solution (7,1)
13
Problem 2
Solve for s by substituting the value for r
into one of the equations.
Subtract the bottom equation from the top
equation.
(-)
Solve for r.
Solution (5,-5)
14
Problem 3
The sum of two numbers is 85. The difference
between the two numbers is 19. What are the two
numbers?
Write an equation for the sum of the
numbers. Write an equation for the difference of
the numbers.
Add the equations together.
Solve for x.
15
Problem 3
Solve for y by substituting in the value of x.
The two numbers are 52 and 33.