Finite Math Examples

Solve by Substitution x+y-z=-5 , 4x-y+z=-5 , x-3y+2z=-10
, ,
Step 1
Move all terms not containing to the right side of the equation.
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Step 1.1
Subtract from both sides of the equation.
Step 1.2
Add to both sides of the equation.
Step 2
Replace all occurrences of with in each equation.
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Step 2.1
Replace all occurrences of in with .
Step 2.2
Simplify the left side.
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Step 2.2.1
Simplify .
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Step 2.2.1.1
Simplify each term.
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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 2.2.1.2
Simplify by adding terms.
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Step 2.2.1.2.1
Subtract from .
Step 2.2.1.2.2
Add and .
Step 2.3
Replace all occurrences of in with .
Step 2.4
Simplify the left side.
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Step 2.4.1
Simplify .
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Step 2.4.1.1
Subtract from .
Step 2.4.1.2
Add and .
Step 3
Solve for in .
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Step 3.1
Move all terms not containing to the right side of the equation.
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Step 3.1.1
Add to both sides of the equation.
Step 3.1.2
Subtract from both sides of the equation.
Step 3.1.3
Add and .
Step 3.2
Divide each term in by and simplify.
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Step 3.2.1
Divide each term in by .
Step 3.2.2
Simplify the left side.
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Step 3.2.2.1
Cancel the common factor of .
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Step 3.2.2.1.1
Cancel the common factor.
Step 3.2.2.1.2
Divide by .
Step 3.2.3
Simplify the right side.
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Step 3.2.3.1
Simplify each term.
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Step 3.2.3.1.1
Dividing two negative values results in a positive value.
Step 3.2.3.1.2
Dividing two negative values results in a positive value.
Step 4
Replace all occurrences of with in each equation.
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Step 4.1
Replace all occurrences of in with .
Step 4.2
Simplify the left side.
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Step 4.2.1
Simplify .
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Step 4.2.1.1
Simplify each term.
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Step 4.2.1.1.1
Apply the distributive property.
Step 4.2.1.1.2
Multiply .
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Step 4.2.1.1.2.1
Combine and .
Step 4.2.1.1.2.2
Multiply by .
Step 4.2.1.1.3
Multiply .
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Step 4.2.1.1.3.1
Combine and .
Step 4.2.1.1.3.2
Multiply by .
Step 4.2.1.1.4
Simplify each term.
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Step 4.2.1.1.4.1
Move the negative in front of the fraction.
Step 4.2.1.1.4.2
Move the negative in front of the fraction.
Step 4.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 4.2.1.3
Combine and .
Step 4.2.1.4
Combine the numerators over the common denominator.
Step 4.2.1.5
Simplify the numerator.
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Step 4.2.1.5.1
Multiply by .
Step 4.2.1.5.2
Subtract from .
Step 4.2.1.6
Move the negative in front of the fraction.
Step 4.2.1.7
To write as a fraction with a common denominator, multiply by .
Step 4.2.1.8
Combine and .
Step 4.2.1.9
Combine the numerators over the common denominator.
Step 4.2.1.10
Combine the numerators over the common denominator.
Step 4.2.1.11
Multiply by .
Step 4.2.1.12
Add and .
Step 4.2.1.13
Factor out of .
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Step 4.2.1.13.1
Factor out of .
Step 4.2.1.13.2
Factor out of .
Step 4.2.1.14
Rewrite as .
Step 4.2.1.15
Factor out of .
Step 4.2.1.16
Factor out of .
Step 4.2.1.17
Move the negative in front of the fraction.
Step 4.3
Replace all occurrences of in with .
Step 4.4
Simplify the right side.
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Step 4.4.1
Simplify .
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Step 4.4.1.1
Apply the distributive property.
Step 4.4.1.2
To write as a fraction with a common denominator, multiply by .
Step 4.4.1.3
Combine and .
Step 4.4.1.4
Combine the numerators over the common denominator.
Step 4.4.1.5
Simplify the numerator.
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Step 4.4.1.5.1
Multiply by .
Step 4.4.1.5.2
Subtract from .
Step 4.4.1.6
Move the negative in front of the fraction.
Step 4.4.1.7
To write as a fraction with a common denominator, multiply by .
Step 4.4.1.8
Simplify terms.
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Step 4.4.1.8.1
Combine and .
Step 4.4.1.8.2
Combine the numerators over the common denominator.
Step 4.4.1.8.3
Combine the numerators over the common denominator.
Step 4.4.1.9
Move to the left of .
Step 4.4.1.10
Simplify terms.
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Step 4.4.1.10.1
Add and .
Step 4.4.1.10.2
Rewrite as .
Step 4.4.1.10.3
Factor out of .
Step 4.4.1.10.4
Factor out of .
Step 4.4.1.10.5
Move the negative in front of the fraction.
Step 5
Solve for in .
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Step 5.1
Multiply both sides of the equation by .
Step 5.2
Simplify both sides of the equation.
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Step 5.2.1
Simplify the left side.
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Step 5.2.1.1
Simplify .
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Step 5.2.1.1.1
Cancel the common factor of .
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Step 5.2.1.1.1.1
Move the leading negative in into the numerator.
Step 5.2.1.1.1.2
Move the leading negative in into the numerator.
Step 5.2.1.1.1.3
Factor out of .
Step 5.2.1.1.1.4
Cancel the common factor.
Step 5.2.1.1.1.5
Rewrite the expression.
Step 5.2.1.1.2
Cancel the common factor of .
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Step 5.2.1.1.2.1
Factor out of .
Step 5.2.1.1.2.2
Cancel the common factor.
Step 5.2.1.1.2.3
Rewrite the expression.
Step 5.2.1.1.3
Multiply.
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Step 5.2.1.1.3.1
Multiply by .
Step 5.2.1.1.3.2
Multiply by .
Step 5.2.2
Simplify the right side.
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Step 5.2.2.1
Simplify .
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Step 5.2.2.1.1
Cancel the common factor of .
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Step 5.2.2.1.1.1
Move the leading negative in into the numerator.
Step 5.2.2.1.1.2
Factor out of .
Step 5.2.2.1.1.3
Cancel the common factor.
Step 5.2.2.1.1.4
Rewrite the expression.
Step 5.2.2.1.2
Multiply by .
Step 5.3
Move all terms not containing to the right side of the equation.
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Step 5.3.1
Subtract from both sides of the equation.
Step 5.3.2
Subtract from .
Step 5.4
Divide each term in by and simplify.
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Step 5.4.1
Divide each term in by .
Step 5.4.2
Simplify the left side.
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Step 5.4.2.1
Dividing two negative values results in a positive value.
Step 5.4.2.2
Divide by .
Step 5.4.3
Simplify the right side.
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Step 5.4.3.1
Divide by .
Step 6
Replace all occurrences of with in each equation.
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Step 6.1
Replace all occurrences of in with .
Step 6.2
Simplify the right side.
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Step 6.2.1
Simplify .
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Step 6.2.1.1
Simplify the numerator.
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Step 6.2.1.1.1
Multiply by .
Step 6.2.1.1.2
Subtract from .
Step 6.2.1.2
Simplify the expression.
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Step 6.2.1.2.1
Divide by .
Step 6.2.1.2.2
Multiply by .
Step 6.3
Replace all occurrences of in with .
Step 6.4
Simplify the right side.
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Step 6.4.1
Simplify .
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Step 6.4.1.1
Combine the numerators over the common denominator.
Step 6.4.1.2
Simplify the expression.
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Step 6.4.1.2.1
Multiply by .
Step 6.4.1.2.2
Add and .
Step 6.4.1.2.3
Divide by .
Step 7
The solution to the system is the complete set of ordered pairs that are valid solutions.
Step 8
The result can be shown in multiple forms.
Point Form:
Equation Form: