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Elimination

Using Multiplication

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.

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.

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.

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).

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.

Summary of steps

- Arrange the equations with like terms in columns.
- Multiply one or both equations by an appropriate

factor so that the new coefficients of x or y

have the same absolute value. - Add or subtract the equations and solve for the

remaining variable. - Substitute the value for that variable into one

of the equations and solve for the value of the

other variable. - Check the solution in each of the original

equations.

You Try It

Solve the following systems of equations by using

elimination.

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.

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).

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.

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.