Finite Math Examples

Solve by Substitution 3x-9y+0z=1 , 3x+9y-9z=3 , 2x+0y+5z=0
, ,
Step 1
Solve for in .
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Step 1.1
Add to both sides of the equation.
Step 1.2
Divide each term in by and simplify.
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Step 1.2.1
Divide each term in by .
Step 1.2.2
Simplify the left side.
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Step 1.2.2.1
Cancel the common factor of .
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Step 1.2.2.1.1
Cancel the common factor.
Step 1.2.2.1.2
Divide by .
Step 1.2.3
Simplify the right side.
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Step 1.2.3.1
Cancel the common factor of and .
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Step 1.2.3.1.1
Factor out of .
Step 1.2.3.1.2
Cancel the common factors.
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Step 1.2.3.1.2.1
Factor out of .
Step 1.2.3.1.2.2
Cancel the common factor.
Step 1.2.3.1.2.3
Rewrite the expression.
Step 1.2.3.1.2.4
Divide by .
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
Cancel the common factor of .
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Step 2.2.1.1.2.1
Cancel the common factor.
Step 2.2.1.1.2.2
Rewrite the expression.
Step 2.2.1.1.3
Multiply by .
Step 2.2.1.2
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 each term.
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Step 2.4.1.1
Apply the distributive property.
Step 2.4.1.2
Combine and .
Step 2.4.1.3
Multiply by .
Step 3
Reorder and .
Step 4
Solve for in .
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Step 4.1
Move all terms not containing to the right side of the equation.
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Step 4.1.1
Subtract from both sides of the equation.
Step 4.1.2
Subtract from both sides of the equation.
Step 4.2
Divide each term in by and simplify.
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Step 4.2.1
Divide each term in by .
Step 4.2.2
Simplify the left side.
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Step 4.2.2.1
Cancel the common factor of .
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Step 4.2.2.1.1
Cancel the common factor.
Step 4.2.2.1.2
Divide by .
Step 4.2.3
Simplify the right side.
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Step 4.2.3.1
Simplify each term.
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Step 4.2.3.1.1
Multiply the numerator by the reciprocal of the denominator.
Step 4.2.3.1.2
Cancel the common factor of .
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Step 4.2.3.1.2.1
Move the leading negative in into the numerator.
Step 4.2.3.1.2.2
Factor out of .
Step 4.2.3.1.2.3
Factor out of .
Step 4.2.3.1.2.4
Cancel the common factor.
Step 4.2.3.1.2.5
Rewrite the expression.
Step 4.2.3.1.3
Multiply by .
Step 4.2.3.1.4
Multiply by .
Step 4.2.3.1.5
Move the negative in front of the fraction.
Step 4.2.3.1.6
Move the negative in front of the fraction.
Step 5
Replace all occurrences of with in each equation.
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Step 5.1
Replace all occurrences of in with .
Step 5.2
Simplify the right side.
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Step 5.2.1
Simplify .
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Step 5.2.1.1
Simplify each term.
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Step 5.2.1.1.1
Apply the distributive property.
Step 5.2.1.1.2
Cancel the common factor of .
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Step 5.2.1.1.2.1
Move the leading negative in into the numerator.
Step 5.2.1.1.2.2
Factor out of .
Step 5.2.1.1.2.3
Cancel the common factor.
Step 5.2.1.1.2.4
Rewrite the expression.
Step 5.2.1.1.3
Cancel the common factor of .
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Step 5.2.1.1.3.1
Move the leading negative in into the numerator.
Step 5.2.1.1.3.2
Factor out of .
Step 5.2.1.1.3.3
Cancel the common factor.
Step 5.2.1.1.3.4
Rewrite the expression.
Step 5.2.1.1.4
Simplify each term.
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Step 5.2.1.1.4.1
Move the negative in front of the fraction.
Step 5.2.1.1.4.2
Move the negative in front of the fraction.
Step 5.2.1.2
Combine the opposite terms in .
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Step 5.2.1.2.1
Add and .
Step 5.2.1.2.2
Add and .
Step 5.3
Replace all occurrences of in with .
Step 5.4
Simplify the left side.
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Step 5.4.1
Simplify .
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Step 5.4.1.1
Simplify each term.
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Step 5.4.1.1.1
Apply the distributive property.
Step 5.4.1.1.2
Cancel the common factor of .
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Step 5.4.1.1.2.1
Move the leading negative in into the numerator.
Step 5.4.1.1.2.2
Factor out of .
Step 5.4.1.1.2.3
Cancel the common factor.
Step 5.4.1.1.2.4
Rewrite the expression.
Step 5.4.1.1.3
Multiply by .
Step 5.4.1.1.4
Cancel the common factor of .
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Step 5.4.1.1.4.1
Move the leading negative in into the numerator.
Step 5.4.1.1.4.2
Factor out of .
Step 5.4.1.1.4.3
Cancel the common factor.
Step 5.4.1.1.4.4
Rewrite the expression.
Step 5.4.1.1.5
Multiply by .
Step 5.4.1.2
Simplify by adding terms.
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Step 5.4.1.2.1
Subtract from .
Step 5.4.1.2.2
Subtract from .
Step 6
Solve for in .
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Step 6.1
Move all terms not containing to the right side of the equation.
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Step 6.1.1
Add to both sides of the equation.
Step 6.1.2
Add and .
Step 6.2
Divide each term in by and simplify.
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Step 6.2.1
Divide each term in by .
Step 6.2.2
Simplify the left side.
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Step 6.2.2.1
Cancel the common factor of .
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Step 6.2.2.1.1
Cancel the common factor.
Step 6.2.2.1.2
Divide by .
Step 6.2.3
Simplify the right side.
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Step 6.2.3.1
Cancel the common factor of and .
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Step 6.2.3.1.1
Factor out of .
Step 6.2.3.1.2
Cancel the common factors.
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Step 6.2.3.1.2.1
Factor out of .
Step 6.2.3.1.2.2
Cancel the common factor.
Step 6.2.3.1.2.3
Rewrite the expression.
Step 6.2.3.2
Move the negative in front of the fraction.
Step 7
Replace all occurrences of with in each equation.
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Step 7.1
Replace all occurrences of in with .
Step 7.2
Simplify the right side.
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Step 7.2.1
Simplify .
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Step 7.2.1.1
Simplify the numerator.
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Step 7.2.1.1.1
Multiply by .
Step 7.2.1.1.2
Combine and .
Step 7.2.1.2
Move the negative in front of the fraction.
Step 7.2.1.3
Multiply the numerator by the reciprocal of the denominator.
Step 7.2.1.4
Multiply .
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Step 7.2.1.4.1
Multiply by .
Step 7.2.1.4.2
Multiply by .
Step 7.2.1.5
Multiply .
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Step 7.2.1.5.1
Multiply by .
Step 7.2.1.5.2
Multiply by .
Step 7.3
Replace all occurrences of in with .
Step 7.4
Simplify the right side.
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Step 7.4.1
Simplify .
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Step 7.4.1.1
Simplify each term.
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Step 7.4.1.1.1
Simplify the numerator.
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Step 7.4.1.1.1.1
Multiply by .
Step 7.4.1.1.1.2
Combine and .
Step 7.4.1.1.2
Move the negative in front of the fraction.
Step 7.4.1.1.3
Multiply the numerator by the reciprocal of the denominator.
Step 7.4.1.1.4
Multiply .
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Step 7.4.1.1.4.1
Multiply by .
Step 7.4.1.1.4.2
Multiply by .
Step 7.4.1.1.5
Multiply .
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Step 7.4.1.1.5.1
Multiply by .
Step 7.4.1.1.5.2
Multiply by .
Step 7.4.1.2
To write as a fraction with a common denominator, multiply by .
Step 7.4.1.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 7.4.1.3.1
Multiply by .
Step 7.4.1.3.2
Multiply by .
Step 7.4.1.4
Combine the numerators over the common denominator.
Step 7.4.1.5
Add and .
Step 8
The solution to the system is the complete set of ordered pairs that are valid solutions.
Step 9
The result can be shown in multiple forms.
Point Form:
Equation Form: