If the coefficients of both variables are different in each equation, the equations may each need to be multiplied by some value to make one of the variable's coefficients match. The equations may also need to be rearranged into the same format before they are added or subtracted.

Before continuing the lesson, consider this system of equations:

«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«mi»x«/mi»«mo»-«/mo»«mn»18«/mn»«mo»=«/mo»«mn»3«/mn»«mi»y«/mi»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»8«/mn»«mi»y«/mi»«mo»+«/mo»«mn»3«/mn»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«/math»

Try the following:

Note: There are many ways to rearrange the equations so that they match. This solution just shows one possibility.
 Equation 1
 Equation 2
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«mi»x«/mi»«mo»-«/mo»«mn»18«/mn»«mo»=«/mo»«mn»3«/mn»«mi»y«/mi»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«mi»x«/mi»«menclose notation=¨updiagonalstrike¨»«mo»-«/mo»«mn»18«/mn»«mo»§#160;«/mo»«/menclose»«menclose mathcolor=¨#1C82BA¨ notation=¨updiagonalstrike¨»«mo»+«/mo»«mn»18«/mn»«/menclose»«mo»=«/mo»«mn»3«/mn»«mi»y«/mi»«mo mathcolor=¨#1C82BA¨»+«/mo»«mn mathcolor=¨#1C82BA¨»18«/mn»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#191919¨»5«/mn»«mi mathcolor=¨#191919¨»x«/mi»«mo mathcolor=¨#191919¨»=«/mo»«mn mathcolor=¨#191919¨»3«/mn»«mi mathcolor=¨#191919¨»y«/mi»«mo mathcolor=¨#191919¨»+«/mo»«mn mathcolor=¨#191919¨»18«/mn»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#191919¨»5«/mn»«mi mathcolor=¨#191919¨»x«/mi»«mo mathcolor=¨#1C82BA¨»-«/mo»«mn mathcolor=¨#1C82BA¨»3«/mn»«mi mathcolor=¨#1C82BA¨»y«/mi»«mo mathcolor=¨#191919¨»=«/mo»«menclose mathcolor=¨#191919¨ notation=¨updiagonalstrike¨»«mn»3«/mn»«mi»y«/mi»«/menclose»«mo mathcolor=¨#191919¨»+«/mo»«mn mathcolor=¨#191919¨»18«/mn»«menclose mathcolor=¨#1C82BA¨ notation=¨updiagonalstrike¨»«mo»-«/mo»«mn»3«/mn»«mi»y«/mi»«/menclose»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»5«/mn»«mi mathvariant=¨bold-italic¨ mathcolor=¨#191919¨»x«/mi»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»3«/mn»«mi mathvariant=¨bold-italic¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»18«/mn»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»8«/mn»«mi»y«/mi»«mo»+«/mo»«mn»3«/mn»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»8«/mn»«mi»y«/mi»«menclose notation=¨updiagonalstrike¨»«mo»+«/mo»«mn»3«/mn»«/menclose»«menclose mathcolor=¨#1C82BA¨ notation=¨updiagonalstrike¨»«mo»-«/mo»«mn»3«/mn»«/menclose»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«mo mathcolor=¨#1C82BA¨»-«/mo»«mn mathcolor=¨#1C82BA¨»3«/mn»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»8«/mn»«mi»y«/mi»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«mn»3«/mn»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»8«/mn»«mi»y«/mi»«mo mathcolor=¨#1C82BA¨»-«/mo»«mn mathcolor=¨#1C82BA¨»3«/mn»«mi mathcolor=¨#1C82BA¨»x«/mi»«mo»=«/mo»«menclose notation=¨updiagonalstrike¨»«mn»3«/mn»«mi»x«/mi»«/menclose»«mo»-«/mo»«mn»3«/mn»«menclose mathcolor=¨#1C82BA¨ notation=¨updiagonalstrike¨»«mo»-«/mo»«mn»3«/mn»«mi»x«/mi»«/menclose»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»3«/mn»«mi mathvariant=¨bold-italic¨»x«/mi»«mo mathvariant=¨bold¨»+«/mo»«mn mathvariant=¨bold¨»8«/mn»«mi mathvariant=¨bold-italic¨»y«/mi»«mo mathvariant=¨bold¨»=«/mo»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»3«/mn»«/math»


Multiply the first equation by 3 and the second equation by 5.

 Equation 1
 Equation 2
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#364868¨»3«/mn»«mfenced mathcolor=¨#364868¨»«mrow»«mn»5«/mn»«mi»x«/mi»«mo»-«/mo»«mn»3«/mn»«mi»y«/mi»«/mrow»«/mfenced»«mo mathcolor=¨#364868¨»=«/mo»«mn mathcolor=¨#364868¨»3«/mn»«mfenced mathcolor=¨#364868¨»«mn»18«/mn»«/mfenced»«/math» 
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathvariant=¨bold¨ mathcolor=¨#364868¨»15«/mn»«mi mathvariant=¨bold-italic¨ mathcolor=¨#364868¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#364868¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#364868¨»9«/mn»«mi mathvariant=¨bold-italic¨ mathcolor=¨#364868¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#364868¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#364868¨»54«/mn»«/math»
 «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#364868¨»5«/mn»«mo mathcolor=¨#364868¨»(«/mo»«mo mathcolor=¨#364868¨»-«/mo»«mn mathcolor=¨#364868¨»3«/mn»«mi mathcolor=¨#364868¨»x«/mi»«mo mathcolor=¨#364868¨»+«/mo»«mn mathcolor=¨#364868¨»8«/mn»«mi mathcolor=¨#364868¨»y«/mi»«mo mathcolor=¨#364868¨»)«/mo»«mo mathcolor=¨#364868¨»=«/mo»«mn mathcolor=¨#364868¨»5«/mn»«mo mathcolor=¨#364868¨»(«/mo»«mo mathcolor=¨#364868¨»-«/mo»«mn mathcolor=¨#364868¨»3«/mn»«mo mathcolor=¨#364868¨»)«/mo»«mspace linebreak=¨newline¨/»«mo mathvariant=¨bold¨ mathcolor=¨#364868¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#364868¨»15«/mn»«mi mathvariant=¨bold-italic¨ mathcolor=¨#364868¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#364868¨»+«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#364868¨»40«/mn»«mi mathvariant=¨bold-italic¨ mathcolor=¨#364868¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#364868¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#364868¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#364868¨»15«/mn»«/math» 

Note: Even though the coefficients of «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#364868¨»x«/mi»«/math» have opposite signs, we have accomplished the goal because the equations can be added together to eliminate «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#364868¨»x«/mi»«/math».


Multiply the first equation by 8 and the second equation by 3.

 Equation 1
 Equation 2
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#364868¨»8«/mn»«mfenced mathcolor=¨#364868¨»«mrow»«mn»5«/mn»«mi»x«/mi»«mo»-«/mo»«mn»3«/mn»«mi»y«/mi»«/mrow»«/mfenced»«mo mathcolor=¨#364868¨»=«/mo»«mn mathcolor=¨#364868¨»8«/mn»«mfenced mathcolor=¨#364868¨»«mn»18«/mn»«/mfenced»«/math» 
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathvariant=¨bold¨ mathcolor=¨#364868¨»40«/mn»«mi mathvariant=¨bold-italic¨ mathcolor=¨#364868¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#364868¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#364868¨»24«/mn»«mi mathvariant=¨bold-italic¨ mathcolor=¨#364868¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#364868¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#364868¨»144«/mn»«/math»
 «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#364868¨»3«/mn»«mo mathcolor=¨#364868¨»(«/mo»«mo mathcolor=¨#364868¨»-«/mo»«mn mathcolor=¨#364868¨»3«/mn»«mi mathcolor=¨#364868¨»x«/mi»«mo mathcolor=¨#364868¨»+«/mo»«mn mathcolor=¨#364868¨»8«/mn»«mi mathcolor=¨#364868¨»y«/mi»«mo mathcolor=¨#364868¨»)«/mo»«mo mathcolor=¨#364868¨»=«/mo»«mn mathcolor=¨#364868¨»3«/mn»«mo mathcolor=¨#364868¨»(«/mo»«mo mathcolor=¨#364868¨»-«/mo»«mn mathcolor=¨#364868¨»3«/mn»«mo mathcolor=¨#364868¨»)«/mo»«mspace linebreak=¨newline¨/»«mo mathvariant=¨bold¨ mathcolor=¨#364868¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#364868¨»9«/mn»«mi mathvariant=¨bold-italic¨ mathcolor=¨#364868¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#364868¨»+«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#364868¨»24«/mn»«mi mathvariant=¨bold-italic¨ mathcolor=¨#364868¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#364868¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#364868¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#364868¨»9«/mn»«/math» 

Note: Even though the coefficients of «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#364868¨»y«/mi»«/math» have opposite signs, we have accomplished the goal because the equations can be added together to eliminate «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#364868¨»y«/mi»«/math».


Example 2

Solve the following system of equations by elimination.


Begin by rearranging the equations into the same format.

The equations are now in the same format, but neither variable has the same coefficient in each equation. Multiply one or both equations by a constant to give one variable the same coefficient in each equation. Determining a Least Common Multiple (LCM) may help decide what constants to multiply by. In this case, the coefficients of x have an LCM of 14. Multiply both equations by a constant, so the x-coefficients are each 14.

Now the two x-variables have the same coefficient. Subtract the two equations to eliminate the x-variable.

Substitute this known value into either of the original equations to determine the x-value.

The solution is (–4, 6).

Verify the solution.