1. Solving Linear Systems
The Elimination/Linear Combinations Method

2. Introduction
John rented a car and drove 125 miles on a 2 day trip and was charged $ He drove 350 miles on a different 4 day trip and was charged $ for the same kind of rental car.
Find the daily fee and cost per mile
Define the variables:
Let “d” represent the cost per day and “m” represent the cost per mile.
Set up the system

3. Using Substitution to Investigate the Combinations Method
Compare the values in the original system to the values in the 3rd step

4. In Order to Eliminate…
In order to eliminate a value ( get a solution of zero) when we “combine” it with another value the two values must be __________
Opposites
The opposite of 4 is
-2 is
-3/4 is
2x is
-4y is -4 2
3/4
-2x 4y

5. The Elimination Method
Looking for an opposite variable
Combine the equations (Add them together)
Solve for the remaining variable
Substitute the value back into the easiest equation to solve

6. Solve the following systems using the elimination method1. 2. 3.
We eliminate the ____ X
We eliminate the ____ Y
We eliminate the ____ X 6 12 7 14 5 15
____Y = ____
Y = ____
____X = ____
X = ____
____Y = ____
Y = ____ 2 2 3
Substitute the Y value back into the equation where the math is__________
easiest
3x + 2(___) = 7
3x + ____ = 7
3x = ___
X = ___ 2 4 3 1

7. The Elimination Method
Look for variables that are opposite…
and if there aren’t opposites then
Multiply one of the equations by a value to get an opposite variable
Combine the equations
Solve for the remaining variable
Substitute the value back into the easiest equation to solve
Multiply by (-1)

8. The Elimination Method
Look for variables that are opposite
Multiply one of the equations by a value to get an opposite variable
Combine the equations
Solve for the remaining variable
Substitute the value back into the easiest equation to solve
Multiply by (2)

9. Solve the following systems using the elimination method4. 5. 6.
Multiply the _______________
by ____to eliminate_____
Multiply the _______________
by ____to eliminate_____
Multiply the _______________
by ____to eliminate_____
1st equation
2nd equation
2nd equation 2 -1 -2 x p y 1
____Y = ____
Y = ____
____r = ____
r = ____ 5 25
____X = ____
X = ____ 8
-16 5 -2
p + 4(___) = 23
p + ____ = 23
p = ___ 5 20 3

10. The Elimination Method
Look for variables that are opposite
Multiply both of the equations by a value to get an opposite variable
Combine the equations
Solve for the remaining variable
Substitute the value back into the easiest equation to solve
Multiply by 2
Multiply by -3

11. Solve the following systems using the elimination method7. 8. 9.
Multiply the __________ by___
And the___________
by____
Multiply the __________ by___
And the___________
by____
Multiply the __________ by___
And the___________
by____
1st equation -7
1st equation 3
1st equation -4
2nd equation
2nd equation
2nd equation 2 2 3
-29 29
___ Y = ____
Y = ____ 7 28
____X = ____
X = ____
____Y = ____
Y = ____ -2 4 -1 4 -2
2x + 5(___) = 26
2x + ____ = 26
2x = ___
x = ___ 4 20 6 3

12. Elimination Gotcha Solve the system using elimination
Make sure both equations are written in same form. (Usually with both x and y variables on the left side)
Then solve as you would any other system.
Multiply both of the equations by a value to get an opposite variable
Combine the equations
Solve for the remaining variable
Substitute the value back into the easiest equation to solve

13. Car Rental Problem
The car was rented for $26 per day and $0.35 per mile

14. Choosing a method Example Method Why
The value of y is known and can be easily substituted into the other equation
5y and -5y are opposites and are easily eliminated.
B can be easily eliminated by multiplying the first equation by -2
Substitution
Linear Combinations