Elimination Using Multiplication
This idea of solving a system of equations by using
multiplication can be shown using a four-step procedure. Consider
the following system of equations.
3x + 2y = 12
2x + 4y = 16
Step 1 Multiply one equation by a number so
that a variable in the new equation has a coeffient that is the
same as or opposite of the coeffient of the variable in the other
equation.
Multiply the first equation by 2. Then the coefficient of y is
4, which is the same as the coefficient of y in the second
equation. The new first equation is 6x + 4y = 24.
Step 2 Subtract one equation from the other
or add the equations to obtain an equation in which one of the
variables does not appear. Solve for the remaining variable.
| 6x + 4y = 24 |
24 Subtract the second
equation from the new first equation. |
| ( - ) 2x + 4y = 16 |
|
| 4x + 0y = 8 |
|
| 4x = 8 |
|
| x = 2 |
Divide each side by 4. |
Step 3 Substitute the value obtained in Step
2 into one of the original equations.
| 3x + 2y = 12 |
The original equation |
| 3(2) + 2y = 12 |
Substitute 2 for x. |
| 6 + 2y = 12 |
|
Step 4 Solve the resulting equation for the
other variable.
| 6 + 2y = 12 |
|
| 2y = 6 |
Subtract 6 from each side. |
| y = 3 |
Divide each side by 2. |
So, the solution is (2, 3).
It is not necessary that you memorize this procedure, but
having it in mind will clarify what needs to be done when solving
systems of equations. There is more than one way to solve the
system. For instance, one might choose the other variable to
eliminate, or choose to multiply the first equation by -2 and
then add the equations.
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