Examples

Solve by Addition/Elimination
, ,
Step 1
Choose two equations and eliminate one variable. In this case, eliminate .
Step 2
Eliminate from the system.
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Step 2.1
Multiply each equation by the value that makes the coefficients of opposite.
Step 2.2
Simplify.
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Step 2.2.1
Simplify the left side.
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Step 2.2.1.1
Simplify .
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Step 2.2.1.1.1
Apply the distributive property.
Step 2.2.1.1.2
Simplify.
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Step 2.2.1.1.2.1
Multiply by .
Step 2.2.1.1.2.2
Multiply by .
Step 2.2.2
Simplify the right side.
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Step 2.2.2.1
Multiply by .
Step 2.3
Add the two equations together to eliminate from the system.
Step 2.4
The resultant equation has eliminated.
Step 3
Choose another two equations and eliminate .
Step 4
Eliminate from the system.
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Step 4.1
Multiply each equation by the value that makes the coefficients of opposite.
Step 4.2
Simplify.
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Step 4.2.1
Simplify the left side.
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Step 4.2.1.1
Simplify .
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Step 4.2.1.1.1
Apply the distributive property.
Step 4.2.1.1.2
Simplify.
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Step 4.2.1.1.2.1
Multiply by .
Step 4.2.1.1.2.2
Multiply by .
Step 4.2.1.1.2.3
Multiply by .
Step 4.2.2
Simplify the right side.
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Step 4.2.2.1
Multiply by .
Step 4.2.3
Simplify the left side.
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Step 4.2.3.1
Simplify .
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Step 4.2.3.1.1
Apply the distributive property.
Step 4.2.3.1.2
Simplify.
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Step 4.2.3.1.2.1
Multiply by .
Step 4.2.3.1.2.2
Multiply by .
Step 4.2.4
Simplify the right side.
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Step 4.2.4.1
Multiply by .
Step 4.3
Add the two equations together to eliminate from the system.
Step 4.4
The resultant equation has eliminated.
Step 5
Take the resultant equations and eliminate another variable. In this case, eliminate .
Step 6
Eliminate from the system.
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Step 6.1
Multiply each equation by the value that makes the coefficients of opposite.
Step 6.2
Simplify.
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Step 6.2.1
Simplify the left side.
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Step 6.2.1.1
Simplify .
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Step 6.2.1.1.1
Apply the distributive property.
Step 6.2.1.1.2
Multiply.
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Step 6.2.1.1.2.1
Multiply by .
Step 6.2.1.1.2.2
Multiply by .
Step 6.2.2
Simplify the right side.
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Step 6.2.2.1
Multiply by .
Step 6.3
Add the two equations together to eliminate from the system.
Step 6.4
The resultant equation has eliminated.
Step 6.5
Divide each term in by and simplify.
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Step 6.5.1
Divide each term in by .
Step 6.5.2
Simplify the left side.
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Step 6.5.2.1
Cancel the common factor of .
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Step 6.5.2.1.1
Cancel the common factor.
Step 6.5.2.1.2
Divide by .
Step 6.5.3
Simplify the right side.
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Step 6.5.3.1
Cancel the common factor of and .
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Step 6.5.3.1.1
Factor out of .
Step 6.5.3.1.2
Cancel the common factors.
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Step 6.5.3.1.2.1
Factor out of .
Step 6.5.3.1.2.2
Cancel the common factor.
Step 6.5.3.1.2.3
Rewrite the expression.
Step 7
Substitute the value of into an equation with eliminated already and solve for the remaining variable.
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Step 7.1
Substitute the value of into an equation with eliminated already.
Step 7.2
Solve for .
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Step 7.2.1
Simplify each term.
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Step 7.2.1.1
Cancel the common factor of .
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Step 7.2.1.1.1
Factor out of .
Step 7.2.1.1.2
Factor out of .
Step 7.2.1.1.3
Cancel the common factor.
Step 7.2.1.1.4
Rewrite the expression.
Step 7.2.1.2
Combine and .
Step 7.2.1.3
Multiply by .
Step 7.2.2
Move all terms not containing to the right side of the equation.
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Step 7.2.2.1
Subtract from both sides of the equation.
Step 7.2.2.2
To write as a fraction with a common denominator, multiply by .
Step 7.2.2.3
Combine and .
Step 7.2.2.4
Combine the numerators over the common denominator.
Step 7.2.2.5
Simplify the numerator.
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Step 7.2.2.5.1
Multiply by .
Step 7.2.2.5.2
Subtract from .
Step 7.2.2.6
Move the negative in front of the fraction.
Step 7.2.3
Divide each term in by and simplify.
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Step 7.2.3.1
Divide each term in by .
Step 7.2.3.2
Simplify the left side.
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Step 7.2.3.2.1
Cancel the common factor of .
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Step 7.2.3.2.1.1
Cancel the common factor.
Step 7.2.3.2.1.2
Divide by .
Step 7.2.3.3
Simplify the right side.
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Step 7.2.3.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 7.2.3.3.2
Cancel the common factor of .
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Step 7.2.3.3.2.1
Move the leading negative in into the numerator.
Step 7.2.3.3.2.2
Factor out of .
Step 7.2.3.3.2.3
Factor out of .
Step 7.2.3.3.2.4
Cancel the common factor.
Step 7.2.3.3.2.5
Rewrite the expression.
Step 7.2.3.3.3
Multiply by .
Step 7.2.3.3.4
Multiply by .
Step 7.2.3.3.5
Dividing two negative values results in a positive value.
Step 8
Substitute the value of each known variable into one of the initial equations and solve for the last variable.
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Step 8.1
Substitute the value of each known variable into one of the initial equations.
Step 8.2
Solve for .
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Step 8.2.1
Simplify .
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Step 8.2.1.1
Simplify each term.
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Step 8.2.1.1.1
Cancel the common factor of .
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Step 8.2.1.1.1.1
Cancel the common factor.
Step 8.2.1.1.1.2
Rewrite the expression.
Step 8.2.1.1.2
Cancel the common factor of .
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Step 8.2.1.1.2.1
Factor out of .
Step 8.2.1.1.2.2
Cancel the common factor.
Step 8.2.1.1.2.3
Rewrite the expression.
Step 8.2.1.2
Subtract from .
Step 8.2.2
Subtract from both sides of the equation.
Step 9
The solution to the system of equations can be represented as a point.
Step 10
The result can be shown in multiple forms.
Point Form:
Equation Form:
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