Finite Math Examples

Solve by Substitution 2x-y+3z=15 , 4x+y-z=-7 , -3x+4y+5z=8
, ,
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 2.2.1.2
Simplify by adding terms.
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Step 2.2.1.2.1
Add and .
Step 2.2.1.2.2
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 .
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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 by .
Step 2.4.1.2
Simplify by adding terms.
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Step 2.4.1.2.1
Subtract from .
Step 2.4.1.2.2
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
Add to both sides of the equation.
Step 3.1.2
Subtract from both sides of the equation.
Step 3.1.3
Add and .
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
Move the negative in front of the fraction.
Step 3.2.3.1.2
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 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
Multiply by .
Step 4.2.1.1.2.2
Combine and .
Step 4.2.1.1.2.3
Multiply by .
Step 4.2.1.1.3
Multiply .
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Step 4.2.1.1.3.1
Combine and .
Step 4.2.1.1.3.2
Multiply by .
Step 4.2.1.1.4
Move the negative in front of the fraction.
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
Simplify the numerator.
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Step 4.2.1.5.1
Multiply by .
Step 4.2.1.5.2
Add and .
Step 4.2.1.6
Move the negative in front of the fraction.
Step 4.2.1.7
To write as a fraction with a common denominator, multiply by .
Step 4.2.1.8
Combine and .
Step 4.2.1.9
Combine the numerators over the common denominator.
Step 4.2.1.10
Combine the numerators over the common denominator.
Step 4.2.1.11
Multiply by .
Step 4.2.1.12
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
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
Simplify the numerator.
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Step 4.4.1.5.1
Multiply by .
Step 4.4.1.5.2
Add and .
Step 4.4.1.6
To write as a fraction with a common denominator, multiply by .
Step 4.4.1.7
Simplify terms.
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Step 4.4.1.7.1
Combine and .
Step 4.4.1.7.2
Combine the numerators over the common denominator.
Step 4.4.1.7.3
Combine the numerators over the common denominator.
Step 4.4.1.8
Move to the left of .
Step 4.4.1.9
Add and .
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
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
Combine the numerators over the common denominator.
Step 6.4.1.2
Simplify the expression.
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Step 6.4.1.2.1
Multiply by .
Step 6.4.1.2.2
Add and .
Step 6.4.1.2.3
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: