Finite Math Examples

Solve by Substitution x+7y=z+13 , x=5+y-z , x+y+4z=21
, ,
Step 1
Subtract from 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
Remove parentheses.
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
Add and .
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
Subtract from 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
Simplify each term.
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Step 3.2.3.1.1
Cancel the common factor of and .
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Step 3.2.3.1.1.1
Factor out of .
Step 3.2.3.1.1.2
Cancel the common factors.
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Step 3.2.3.1.1.2.1
Factor out of .
Step 3.2.3.1.1.2.2
Cancel the common factor.
Step 3.2.3.1.1.2.3
Rewrite the expression.
Step 3.2.3.1.2
Move the negative in front of the fraction.
Step 3.2.3.1.3
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 .
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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
Remove parentheses.
Step 4.2.1.1.2
Simplify each term.
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Step 4.2.1.1.2.1
Apply the distributive property.
Step 4.2.1.1.2.2
Multiply .
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Step 4.2.1.1.2.2.1
Multiply by .
Step 4.2.1.1.2.2.2
Combine and .
Step 4.2.1.1.2.2.3
Multiply by .
Step 4.2.1.1.2.3
Multiply .
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Step 4.2.1.1.2.3.1
Combine and .
Step 4.2.1.1.2.3.2
Multiply by .
Step 4.2.1.1.2.4
Move the negative in front of the fraction.
Step 4.2.1.1.3
To write as a fraction with a common denominator, multiply by .
Step 4.2.1.1.4
Combine and .
Step 4.2.1.1.5
Combine the numerators over the common denominator.
Step 4.2.1.1.6
Subtract from .
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Step 4.2.1.1.6.1
Reorder and .
Step 4.2.1.1.6.2
Subtract from .
Step 4.2.1.1.7
Move the negative in front of the fraction.
Step 4.2.1.1.8
To write as a fraction with a common denominator, multiply by .
Step 4.2.1.1.9
Combine and .
Step 4.2.1.1.10
Combine the numerators over the common denominator.
Step 4.2.1.1.11
Simplify the numerator.
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Step 4.2.1.1.11.1
Multiply by .
Step 4.2.1.1.11.2
Add and .
Step 4.2.2
Simplify the right side.
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Step 4.2.2.1
Simplify .
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Step 4.2.2.1.1
To write as a fraction with a common denominator, multiply by .
Step 4.2.2.1.2
Combine and .
Step 4.2.2.1.3
Combine the numerators over the common denominator.
Step 4.2.2.1.4
Simplify the numerator.
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Step 4.2.2.1.4.1
Multiply by .
Step 4.2.2.1.4.2
Subtract from .
Step 4.2.2.1.5
To write as a fraction with a common denominator, multiply by .
Step 4.2.2.1.6
Simplify terms.
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Step 4.2.2.1.6.1
Combine and .
Step 4.2.2.1.6.2
Combine the numerators over the common denominator.
Step 4.2.2.1.7
Simplify each term.
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Step 4.2.2.1.7.1
Simplify the numerator.
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Step 4.2.2.1.7.1.1
Factor out of .
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Step 4.2.2.1.7.1.1.1
Factor out of .
Step 4.2.2.1.7.1.1.2
Factor out of .
Step 4.2.2.1.7.1.1.3
Factor out of .
Step 4.2.2.1.7.1.2
Multiply by .
Step 4.2.2.1.7.1.3
Subtract from .
Step 4.2.2.1.7.2
Move to the left of .
Step 4.2.2.1.7.3
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
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
Multiply by .
Step 4.4.1.1.2.2
Combine and .
Step 4.4.1.1.2.3
Multiply by .
Step 4.4.1.1.3
Multiply .
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Step 4.4.1.1.3.1
Combine and .
Step 4.4.1.1.3.2
Multiply by .
Step 4.4.1.1.4
Move the negative in front of the fraction.
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
Subtract from .
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Step 4.4.1.5.1
Reorder and .
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
Combine and .
Step 4.4.1.9
Combine the numerators over the common denominator.
Step 4.4.1.10
Simplify the numerator.
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Step 4.4.1.10.1
Multiply by .
Step 4.4.1.10.2
Add and .
Step 5
Solve for in .
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Step 5.1
Move all terms containing to the left side of the equation.
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Step 5.1.1
Add to both sides of the equation.
Step 5.1.2
Combine the numerators over the common denominator.
Step 5.1.3
Add and .
Step 5.1.4
Simplify each term.
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Step 5.1.4.1
Cancel the common factor of and .
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Step 5.1.4.1.1
Factor out of .
Step 5.1.4.1.2
Cancel the common factors.
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Step 5.1.4.1.2.1
Factor out of .
Step 5.1.4.1.2.2
Cancel the common factor.
Step 5.1.4.1.2.3
Rewrite the expression.
Step 5.1.4.2
Move the negative in front of the fraction.
Step 5.2
Move all terms not containing to the right side of the equation.
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Step 5.2.1
Subtract from both sides of the equation.
Step 5.2.2
Combine the numerators over the common denominator.
Step 5.2.3
Subtract from .
Step 5.2.4
Move the negative in front of the fraction.
Step 5.3
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
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
Cancel the common factor of .
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Step 5.4.2.1.1
Cancel the common factor.
Step 5.4.2.1.2
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 each term.
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Step 6.2.1.1.1
Cancel the common factor of and .
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Step 6.2.1.1.1.1
Factor out of .
Step 6.2.1.1.1.2
Cancel the common factors.
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Step 6.2.1.1.1.2.1
Factor out of .
Step 6.2.1.1.1.2.2
Cancel the common factor.
Step 6.2.1.1.1.2.3
Rewrite the expression.
Step 6.2.1.1.2
Multiply by .
Step 6.2.1.2
Combine fractions.
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Step 6.2.1.2.1
Combine the numerators over the common denominator.
Step 6.2.1.2.2
Simplify the expression.
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Step 6.2.1.2.2.1
Add and .
Step 6.2.1.2.2.2
Divide 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
Multiply by .
Step 6.4.1.2
Cancel the common factor of and .
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Step 6.4.1.2.1
Factor out of .
Step 6.4.1.2.2
Cancel the common factors.
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Step 6.4.1.2.2.1
Factor out of .
Step 6.4.1.2.2.2
Cancel the common factor.
Step 6.4.1.2.2.3
Rewrite the expression.
Step 6.4.1.3
Combine the numerators over the common denominator.
Step 6.4.1.4
Simplify the expression.
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Step 6.4.1.4.1
Add and .
Step 6.4.1.4.2
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: