Finite Math Examples

Solve by Substitution 3x-5y+6z=51 , 5x+4y-z=-7 , 8x+7y+2z=5
, ,
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
Divide by .
Step 1.2.3.1.2
Cancel the common factor of and .
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Step 1.2.3.1.2.1
Factor out of .
Step 1.2.3.1.2.2
Cancel the common factors.
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Step 1.2.3.1.2.2.1
Factor out of .
Step 1.2.3.1.2.2.2
Cancel the common factor.
Step 1.2.3.1.2.2.3
Rewrite the expression.
Step 1.2.3.1.2.2.4
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
Simplify.
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Step 2.2.1.1.2.1
Multiply by .
Step 2.2.1.1.2.2
Multiply .
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Step 2.2.1.1.2.2.1
Combine and .
Step 2.2.1.1.2.2.2
Multiply by .
Step 2.2.1.1.2.3
Multiply by .
Step 2.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 2.2.1.3
Simplify terms.
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Step 2.2.1.3.1
Combine and .
Step 2.2.1.3.2
Combine the numerators over the common denominator.
Step 2.2.1.4
Simplify each term.
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Step 2.2.1.4.1
Simplify the numerator.
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Step 2.2.1.4.1.1
Factor out of .
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Step 2.2.1.4.1.1.1
Factor out of .
Step 2.2.1.4.1.1.2
Factor out of .
Step 2.2.1.4.1.1.3
Factor out of .
Step 2.2.1.4.1.2
Multiply by .
Step 2.2.1.4.1.3
Add and .
Step 2.2.1.4.2
Move to the left of .
Step 2.2.1.5
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
Apply the distributive property.
Step 2.4.1.1.2
Simplify.
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Step 2.4.1.1.2.1
Multiply by .
Step 2.4.1.1.2.2
Multiply .
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Step 2.4.1.1.2.2.1
Combine and .
Step 2.4.1.1.2.2.2
Multiply by .
Step 2.4.1.1.2.3
Multiply by .
Step 2.4.1.2
To write as a fraction with a common denominator, multiply by .
Step 2.4.1.3
Simplify terms.
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Step 2.4.1.3.1
Combine and .
Step 2.4.1.3.2
Combine the numerators over the common denominator.
Step 2.4.1.4
Simplify each term.
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Step 2.4.1.4.1
Simplify the numerator.
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Step 2.4.1.4.1.1
Factor out of .
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Step 2.4.1.4.1.1.1
Factor out of .
Step 2.4.1.4.1.1.2
Factor out of .
Step 2.4.1.4.1.1.3
Factor out of .
Step 2.4.1.4.1.2
Multiply by .
Step 2.4.1.4.1.3
Add and .
Step 2.4.1.4.2
Move to the left of .
Step 2.4.1.5
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
Dividing two negative values results in a positive value.
Step 3.2.3.1.2
Multiply the numerator by the reciprocal of the denominator.
Step 3.2.3.1.3
Move the negative in front of the fraction.
Step 3.2.3.1.4
Multiply .
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Step 3.2.3.1.4.1
Multiply by .
Step 3.2.3.1.4.2
Multiply by .
Step 3.2.3.1.4.3
Multiply by .
Step 3.2.3.1.4.4
Multiply 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
Find the common denominator.
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Step 4.2.1.1.1
Write as a fraction with denominator .
Step 4.2.1.1.2
Multiply by .
Step 4.2.1.1.3
Multiply by .
Step 4.2.1.1.4
Write as a fraction with denominator .
Step 4.2.1.1.5
Multiply by .
Step 4.2.1.1.6
Multiply by .
Step 4.2.1.2
Combine the numerators over the common denominator.
Step 4.2.1.3
Simplify each term.
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Step 4.2.1.3.1
Multiply by .
Step 4.2.1.3.2
Apply the distributive property.
Step 4.2.1.3.3
Multiply .
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Step 4.2.1.3.3.1
Combine and .
Step 4.2.1.3.3.2
Multiply by .
Step 4.2.1.3.4
Multiply .
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Step 4.2.1.3.4.1
Combine and .
Step 4.2.1.3.4.2
Multiply by .
Step 4.2.1.3.5
Simplify each term.
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Step 4.2.1.3.5.1
Move the negative in front of the fraction.
Step 4.2.1.3.5.2
Move the negative in front of the fraction.
Step 4.2.1.3.6
Apply the distributive property.
Step 4.2.1.3.7
Multiply .
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Step 4.2.1.3.7.1
Multiply by .
Step 4.2.1.3.7.2
Combine and .
Step 4.2.1.3.7.3
Multiply by .
Step 4.2.1.3.8
Cancel the common factor of .
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Step 4.2.1.3.8.1
Move the leading negative in into the numerator.
Step 4.2.1.3.8.2
Factor out of .
Step 4.2.1.3.8.3
Cancel the common factor.
Step 4.2.1.3.8.4
Rewrite the expression.
Step 4.2.1.3.9
Simplify each term.
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Step 4.2.1.3.9.1
Move the negative in front of the fraction.
Step 4.2.1.3.9.2
Move the negative in front of the fraction.
Step 4.2.1.4
To write as a fraction with a common denominator, multiply by .
Step 4.2.1.5
Combine and .
Step 4.2.1.6
Combine the numerators over the common denominator.
Step 4.2.1.7
Simplify the numerator.
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Step 4.2.1.7.1
Multiply by .
Step 4.2.1.7.2
Subtract from .
Step 4.2.1.8
Move the negative in front of the fraction.
Step 4.2.1.9
To write as a fraction with a common denominator, multiply by .
Step 4.2.1.10
Combine and .
Step 4.2.1.11
Combine the numerators over the common denominator.
Step 4.2.1.12
Combine the numerators over the common denominator.
Step 4.2.1.13
Multiply by .
Step 4.2.1.14
Add and .
Step 4.2.1.15
Factor out of .
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Step 4.2.1.15.1
Factor out of .
Step 4.2.1.15.2
Factor out of .
Step 4.2.1.15.3
Factor out of .
Step 4.2.1.16
Rewrite as .
Step 4.2.1.17
Factor out of .
Step 4.2.1.18
Factor out of .
Step 4.2.1.19
Move the negative in front of the fraction.
Step 4.2.1.20
Multiply the numerator by the reciprocal of the denominator.
Step 4.2.1.21
Cancel the common factor of .
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Step 4.2.1.21.1
Move the leading negative in into the numerator.
Step 4.2.1.21.2
Factor out of .
Step 4.2.1.21.3
Cancel the common factor.
Step 4.2.1.21.4
Rewrite the expression.
Step 4.2.1.22
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
Cancel the common factor of .
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Step 4.4.1.1.2.1
Factor out of .
Step 4.4.1.1.2.2
Factor out of .
Step 4.4.1.1.2.3
Cancel the common factor.
Step 4.4.1.1.2.4
Rewrite the expression.
Step 4.4.1.1.3
Cancel the common factor of .
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Step 4.4.1.1.3.1
Factor out of .
Step 4.4.1.1.3.2
Factor out of .
Step 4.4.1.1.3.3
Cancel the common factor.
Step 4.4.1.1.3.4
Rewrite the expression.
Step 4.4.1.1.4
Simplify each term.
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Step 4.4.1.1.4.1
Rewrite as .
Step 4.4.1.1.4.2
Rewrite as .
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
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 4.4.1.8.1
Multiply by .
Step 4.4.1.8.2
Multiply by .
Step 4.4.1.9
Combine the numerators over the common denominator.
Step 4.4.1.10
Simplify each term.
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Step 4.4.1.10.1
Simplify the numerator.
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Step 4.4.1.10.1.1
Factor out of .
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Step 4.4.1.10.1.1.1
Factor out of .
Step 4.4.1.10.1.1.2
Factor out of .
Step 4.4.1.10.1.1.3
Factor out of .
Step 4.4.1.10.1.2
Multiply by .
Step 4.4.1.10.1.3
Subtract from .
Step 4.4.1.10.2
Move to the left of .
Step 4.4.1.10.3
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
Factor out of .
Step 5.2.1.1.1.3
Cancel the common factor.
Step 5.2.1.1.1.4
Rewrite the expression.
Step 5.2.1.1.2
Multiply.
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Step 5.2.1.1.2.1
Multiply by .
Step 5.2.1.1.2.2
Multiply by .
Step 5.2.2
Simplify the right side.
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Step 5.2.2.1
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
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 6.2.1.1.3
Move the negative in front of the fraction.
Step 6.2.1.1.4
Multiply .
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Step 6.2.1.1.4.1
Multiply by .
Step 6.2.1.1.4.2
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
Subtract from .
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
Simplify each term.
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Step 6.4.1.2.1
Cancel the common factor of and .
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Step 6.4.1.2.1.1
Factor out of .
Step 6.4.1.2.1.2
Cancel the common factors.
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Step 6.4.1.2.1.2.1
Factor out of .
Step 6.4.1.2.1.2.2
Cancel the common factor.
Step 6.4.1.2.1.2.3
Rewrite the expression.
Step 6.4.1.2.2
Move the negative in front of the fraction.
Step 6.4.1.3
Combine fractions.
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Step 6.4.1.3.1
Combine the numerators over the common denominator.
Step 6.4.1.3.2
Simplify the expression.
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Step 6.4.1.3.2.1
Subtract from .
Step 6.4.1.3.2.2
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: