Elimination

1
Elimination
Using Multiplication
2
Solving by Elimination
Previously, we learned how to solve systems of
equations by using addition or subtraction which
eliminated one of the variables.
This system of equations could be solved by
eliminating the y variable through addition.
This system of equations could be solved by
eliminating the x variable through subtraction.
3
Solving by Elimination
Since this system of equations cant be solved by
elimination with addition or elimination with
subtraction, how can we solve it?
The new system of equations is now
If the top equation was multiplied by 3, then the
first term would be 3x. The bottom equation
could then be subtracted from the top equation
eliminating the variable x.
4
Solving by Elimination
Subtract the bottom equation from the top
equation.
(-)
Solve for y.
Solve for x by substituting the value for y into
one of the equations.
5
Solving by Elimination
Substitute the value of the variables into each
equation to determine if the solution is correct.
?
?
This system of equations represents two lines
which intersect at the point (5,3).
6
Solving by Elimination
Previously, we solved this system of equations by
multiplying the top equation by 3 and then used
elimination by subtraction.
Could we have used a different factor for the
multiplication?
We could have multiplied the bottom equation by 2
to get
The system of equations would then become
Elimination by addition would then be used to
solve this system of equations. The result
should be the same. Try it and see.
7
Summary of steps

1. Arrange the equations with like terms in columns.
2. Multiply one or both equations by an appropriate
   factor so that the new coefficients of x or y
   have the same absolute value.
3. Add or subtract the equations and solve for the
   remaining variable.
4. Substitute the value for that variable into one
   of the equations and solve for the value of the
   other variable.
5. Check the solution in each of the original
   equations.

8
You Try It
Solve the following systems of equations by using
elimination.
9
Problem 1
Multiply the bottom equation by 4 to get a new
system of equations.
(-)
Subtract the bottom equation from the top
equation.
Solve for y.
Solve for x by substituting the value for y into
one of the equations.
10
Problem 1
Check the solution by substituting the values for
the variables into each equation.
?
?
This system of equations represents two lines
which intersect at the point (6,4).
11
Problem 2
The lowest common multiple of 6 and 8 is 24.
Multiply the top equation by 3 and the bottom
equation by 4.
The new system of equations is
Since these two equations are identical, there is
only one line and an infinite number of solutions.
12
Problem 3
The lowest common multiple of 2 and 3 is 6.
Multiply the top equation by 3 and the bottom
equation by 2.
Add the new system of equations together.
Since 0 ? 25, there is no solution to this system
of equations which represents two parallel lines.