Finite Math Examples

Solve by Substitution 3x+2y-1z=2 , 2x-3y+1z=-2 , 1x-1y-z=-4
, ,
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 2.2.1.1.2.3
Multiply by .
Step 2.2.1.2
Simplify by adding terms.
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Step 2.2.1.2.1
Add and .
Step 2.2.1.2.2
Subtract from .
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
Simplify each term.
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Step 2.4.1.1.1
Rewrite as .
Step 2.4.1.1.2
Apply the distributive property.
Step 2.4.1.1.3
Simplify.
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Step 2.4.1.1.3.1
Multiply by .
Step 2.4.1.1.3.2
Multiply by .
Step 2.4.1.1.3.3
Multiply by .
Step 2.4.1.2
Simplify by adding terms.
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Step 2.4.1.2.1
Add and .
Step 2.4.1.2.2
Subtract from .
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
Subtract from .
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
Divide by .
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 by .
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
Find the common denominator.
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Step 4.2.1.5.1
Write as a fraction with denominator .
Step 4.2.1.5.2
Multiply by .
Step 4.2.1.5.3
Multiply by .
Step 4.2.1.5.4
Write as a fraction with denominator .
Step 4.2.1.5.5
Multiply by .
Step 4.2.1.5.6
Multiply by .
Step 4.2.1.6
Combine the numerators over the common denominator.
Step 4.2.1.7
Simplify each term.
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Step 4.2.1.7.1
Multiply by .
Step 4.2.1.7.2
Multiply by .
Step 4.2.1.7.3
Multiply by .
Step 4.2.1.8
Simplify by adding terms.
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Step 4.2.1.8.1
Subtract from .
Step 4.2.1.8.2
Add and .
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
Simplify each term.
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Step 4.4.1.1.1
Apply the distributive property.
Step 4.4.1.1.2
Multiply .
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Step 4.4.1.1.2.1
Combine and .
Step 4.4.1.1.2.2
Multiply by .
Step 4.4.1.1.3
Multiply by .
Step 4.4.1.1.4
Move the negative in front of the fraction.
Step 4.4.1.2
Add and .
Step 4.4.1.3
To write as a fraction with a common denominator, multiply by .
Step 4.4.1.4
Simplify terms.
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Step 4.4.1.4.1
Combine and .
Step 4.4.1.4.2
Combine the numerators over the common denominator.
Step 4.4.1.5
Simplify each term.
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Step 4.4.1.5.1
Simplify the numerator.
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Step 4.4.1.5.1.1
Factor out of .
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Step 4.4.1.5.1.1.1
Factor out of .
Step 4.4.1.5.1.1.2
Factor out of .
Step 4.4.1.5.1.1.3
Factor out of .
Step 4.4.1.5.1.2
Multiply by .
Step 4.4.1.5.1.3
Add and .
Step 4.4.1.5.2
Multiply by .
Step 5
Solve for in .
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Step 5.1
Multiply both sides by .
Step 5.2
Simplify.
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Step 5.2.1
Simplify the left side.
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Step 5.2.1.1
Cancel the common factor of .
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Step 5.2.1.1.1
Cancel the common factor.
Step 5.2.1.1.2
Rewrite the expression.
Step 5.2.2
Simplify the right side.
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Step 5.2.2.1
Multiply by .
Step 5.3
Solve for .
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Step 5.3.1
Move all terms not containing to the right side of the equation.
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Step 5.3.1.1
Add to both sides of the equation.
Step 5.3.1.2
Add and .
Step 5.3.2
Divide each term in by and simplify.
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Step 5.3.2.1
Divide each term in by .
Step 5.3.2.2
Simplify the left side.
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Step 5.3.2.2.1
Cancel the common factor of .
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Step 5.3.2.2.1.1
Cancel the common factor.
Step 5.3.2.2.1.2
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 each term.
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Step 6.2.1.1.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.2.1.1.2
Cancel the common factor of .
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Step 6.2.1.1.2.1
Factor out of .
Step 6.2.1.1.2.2
Cancel the common factor.
Step 6.2.1.1.2.3
Rewrite the expression.
Step 6.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 6.2.1.3
Combine and .
Step 6.2.1.4
Combine the numerators over the common denominator.
Step 6.2.1.5
Simplify the numerator.
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Step 6.2.1.5.1
Multiply by .
Step 6.2.1.5.2
Add and .
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
Simplify each term.
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Step 6.4.1.1.1
Combine and .
Step 6.4.1.1.2
Multiply by .
Step 6.4.1.1.3
Multiply the numerator by the reciprocal of the denominator.
Step 6.4.1.1.4
Cancel the common factor of .
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Step 6.4.1.1.4.1
Factor out of .
Step 6.4.1.1.4.2
Cancel the common factor.
Step 6.4.1.1.4.3
Rewrite the expression.
Step 6.4.1.2
To write as a fraction with a common denominator, multiply by .
Step 6.4.1.3
Combine and .
Step 6.4.1.4
Combine the numerators over the common denominator.
Step 6.4.1.5
Simplify the numerator.
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Step 6.4.1.5.1
Multiply by .
Step 6.4.1.5.2
Subtract from .
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: