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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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