Finite Math Examples

Solve by Substitution -8x-y+3z=28 , -7y+7z=42 , 6x+2y+3z=-9
, ,
Step 1
Solve for in .
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Step 1.1
Move all terms not containing to the right side of the equation.
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Step 1.1.1
Add to both sides of the equation.
Step 1.1.2
Subtract from 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
Simplify each term.
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Step 1.2.3.1.1
Cancel the common factor of and .
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Step 1.2.3.1.1.1
Factor out of .
Step 1.2.3.1.1.2
Cancel the common factors.
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Step 1.2.3.1.1.2.1
Factor out of .
Step 1.2.3.1.1.2.2
Cancel the common factor.
Step 1.2.3.1.1.2.3
Rewrite the expression.
Step 1.2.3.1.2
Move the negative in front of the fraction.
Step 1.2.3.1.3
Move the negative in front of the fraction.
Step 1.2.3.1.4
Dividing two negative values results in a positive value.
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
Cancel the common factor of .
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Step 2.2.1.1.2.1.1
Move the leading negative in into the numerator.
Step 2.2.1.1.2.1.2
Factor out of .
Step 2.2.1.1.2.1.3
Cancel the common factor.
Step 2.2.1.1.2.1.4
Rewrite the expression.
Step 2.2.1.1.2.2
Multiply by .
Step 2.2.1.1.2.3
Cancel the common factor of .
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Step 2.2.1.1.2.3.1
Move the leading negative in into the numerator.
Step 2.2.1.1.2.3.2
Factor out of .
Step 2.2.1.1.2.3.3
Factor out of .
Step 2.2.1.1.2.3.4
Cancel the common factor.
Step 2.2.1.1.2.3.5
Rewrite the expression.
Step 2.2.1.1.2.4
Combine and .
Step 2.2.1.1.2.5
Multiply by .
Step 2.2.1.1.2.6
Cancel the common factor of .
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Step 2.2.1.1.2.6.1
Factor out of .
Step 2.2.1.1.2.6.2
Factor out of .
Step 2.2.1.1.2.6.3
Cancel the common factor.
Step 2.2.1.1.2.6.4
Rewrite the expression.
Step 2.2.1.1.2.7
Combine and .
Step 2.2.1.1.2.8
Multiply by .
Step 2.2.1.1.3
Move the negative in front of the fraction.
Step 2.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 2.2.1.3
Combine and .
Step 2.2.1.4
Combine the numerators over the common denominator.
Step 2.2.1.5
Combine the numerators over the common denominator.
Step 2.2.1.6
Multiply by .
Step 2.2.1.7
Add and .
Step 2.2.1.8
To write as a fraction with a common denominator, multiply by .
Step 2.2.1.9
Combine and .
Step 2.2.1.10
Combine the numerators over the common denominator.
Step 2.2.1.11
Multiply by .
Step 2.2.1.12
To write as a fraction with a common denominator, multiply by .
Step 2.2.1.13
Simplify terms.
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Step 2.2.1.13.1
Combine and .
Step 2.2.1.13.2
Combine the numerators over the common denominator.
Step 2.2.1.14
Simplify the numerator.
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Step 2.2.1.14.1
Multiply by .
Step 2.2.1.14.2
Add and .
Step 3
Solve for in .
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Step 3.1
Subtract from both sides of the equation.
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
Divide by .
Step 3.2.3.1.2
Cancel the common factor of .
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Step 3.2.3.1.2.1
Cancel the common factor.
Step 3.2.3.1.2.2
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 the numerator.
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Step 4.2.1.1.1
Apply the distributive property.
Step 4.2.1.1.2
Multiply by .
Step 4.2.1.1.3
Add and .
Step 4.2.1.1.4
Subtract from .
Step 4.2.1.1.5
Factor out of .
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Step 4.2.1.1.5.1
Factor out of .
Step 4.2.1.1.5.2
Factor out of .
Step 4.2.1.1.5.3
Factor out of .
Step 4.2.1.2
Cancel the common factors.
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Step 4.2.1.2.1
Factor out of .
Step 4.2.1.2.2
Cancel the common factor.
Step 4.2.1.2.3
Rewrite the expression.
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
Combine the numerators over the common denominator.
Step 4.4.1.2
Simplify each term.
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Step 4.4.1.2.1
Apply the distributive property.
Step 4.4.1.2.2
Multiply by .
Step 4.4.1.3
Add and .
Step 4.4.1.4
Simplify each term.
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Step 4.4.1.4.1
Move the negative in front of the fraction.
Step 4.4.1.4.2
Cancel the common factor of and .
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Step 4.4.1.4.2.1
Factor out of .
Step 4.4.1.4.2.2
Factor out of .
Step 4.4.1.4.2.3
Factor out of .
Step 4.4.1.4.2.4
Cancel the common factors.
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Step 4.4.1.4.2.4.1
Factor out of .
Step 4.4.1.4.2.4.2
Cancel the common factor.
Step 4.4.1.4.2.4.3
Rewrite the expression.
Step 4.4.1.5
To write as a fraction with a common denominator, multiply by .
Step 4.4.1.6
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 4.4.1.6.1
Multiply by .
Step 4.4.1.6.2
Multiply by .
Step 4.4.1.7
Combine the numerators over the common denominator.
Step 4.4.1.8
Simplify the numerator.
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Step 4.4.1.8.1
Multiply by .
Step 4.4.1.8.2
Add and .
Step 4.4.1.9
Simplify with factoring out.
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Step 4.4.1.9.1
Rewrite as .
Step 4.4.1.9.2
Factor out of .
Step 4.4.1.9.3
Factor out of .
Step 4.4.1.9.4
Move the negative in front of the fraction.
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 5.3.2.3
Simplify the right side.
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Step 5.3.2.3.1
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 .
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Step 6.4.1
Simplify the left side.
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Step 6.4.1.1
Remove parentheses.
Step 6.4.2
Simplify the right side.
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Step 6.4.2.1
Add and .
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: