Finite Math Examples

Solve by Substitution a+b+c=240 , a/4+b/2+c/5=60 , 2b+5c-810=0
, ,
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
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
Simplify .
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Step 2.2.1.1
To write as a fraction with a common denominator, multiply by .
Step 2.2.1.2
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 2.2.1.2.1
Multiply by .
Step 2.2.1.2.2
Multiply by .
Step 2.2.1.3
Combine the numerators over the common denominator.
Step 2.2.1.4
Add and .
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Step 2.2.1.4.1
Reorder and .
Step 2.2.1.4.2
Add and .
Step 2.2.1.5
To write as a fraction with a common denominator, multiply by .
Step 2.2.1.6
To write as a fraction with a common denominator, multiply by .
Step 2.2.1.7
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 2.2.1.7.1
Multiply by .
Step 2.2.1.7.2
Multiply by .
Step 2.2.1.7.3
Multiply by .
Step 2.2.1.7.4
Multiply by .
Step 2.2.1.8
Combine the numerators over the common denominator.
Step 2.2.1.9
Simplify the numerator.
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Step 2.2.1.9.1
Apply the distributive property.
Step 2.2.1.9.2
Simplify.
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Step 2.2.1.9.2.1
Multiply by .
Step 2.2.1.9.2.2
Move to the left of .
Step 2.2.1.9.2.3
Multiply by .
Step 2.2.1.9.3
Move to the left of .
Step 2.2.1.9.4
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
Add to 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
Move the negative in front of the fraction.
Step 3.2.3.1.2
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 .
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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 by .
Step 4.2.1.1.4
Move the negative in front of the fraction.
Step 4.2.1.1.5
Add and .
Step 4.2.1.1.6
To write as a fraction with a common denominator, multiply by .
Step 4.2.1.1.7
Combine and .
Step 4.2.1.1.8
Combine the numerators over the common denominator.
Step 4.2.1.1.9
To write as a fraction with a common denominator, multiply by .
Step 4.2.1.1.10
Combine and .
Step 4.2.1.1.11
Combine the numerators over the common denominator.
Step 4.2.1.1.12
Rewrite in a factored form.
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Step 4.2.1.1.12.1
Multiply by .
Step 4.2.1.1.12.2
Multiply by .
Step 4.2.1.1.12.3
Subtract from .
Step 4.2.1.1.12.4
Factor out of .
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Step 4.2.1.1.12.4.1
Factor out of .
Step 4.2.1.1.12.4.2
Factor out of .
Step 4.2.1.1.12.4.3
Factor out of .
Step 4.2.1.2
Multiply the numerator by the reciprocal of the denominator.
Step 4.2.1.3
Multiply .
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Step 4.2.1.3.1
Multiply by .
Step 4.2.1.3.2
Multiply by .
Step 4.2.1.4
Factor out of .
Step 4.2.1.5
Rewrite as .
Step 4.2.1.6
Factor out of .
Step 4.2.1.7
Simplify the expression.
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Step 4.2.1.7.1
Rewrite as .
Step 4.2.1.7.2
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
Multiply by .
Step 4.4.1.1.3
Multiply by .
Step 4.4.1.2
Subtract from .
Step 4.4.1.3
To write as a fraction with a common denominator, multiply by .
Step 4.4.1.4
Simplify terms.
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Step 4.4.1.4.1
Combine and .
Step 4.4.1.4.2
Combine the numerators over the common denominator.
Step 4.4.1.5
Simplify each term.
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Step 4.4.1.5.1
Simplify the numerator.
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Step 4.4.1.5.1.1
Factor out of .
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Step 4.4.1.5.1.1.1
Factor out of .
Step 4.4.1.5.1.1.2
Factor out of .
Step 4.4.1.5.1.1.3
Factor out of .
Step 4.4.1.5.1.2
Multiply by .
Step 4.4.1.5.1.3
Subtract from .
Step 4.4.1.5.2
Move to the left of .
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
Move the leading negative in into the numerator.
Step 5.2.1.1.1.3
Factor out of .
Step 5.2.1.1.1.4
Cancel the common factor.
Step 5.2.1.1.1.5
Rewrite the expression.
Step 5.2.1.1.2
Cancel the common factor of .
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Step 5.2.1.1.2.1
Factor out of .
Step 5.2.1.1.2.2
Cancel the common factor.
Step 5.2.1.1.2.3
Rewrite the expression.
Step 5.2.1.1.3
Multiply.
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Step 5.2.1.1.3.1
Multiply by .
Step 5.2.1.1.3.2
Multiply by .
Step 5.2.2
Simplify the right side.
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Step 5.2.2.1
Simplify .
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Step 5.2.2.1.1
Cancel the common factor of .
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Step 5.2.2.1.1.1
Move the leading negative in into the numerator.
Step 5.2.2.1.1.2
Factor out of .
Step 5.2.2.1.1.3
Cancel the common factor.
Step 5.2.2.1.1.4
Rewrite the expression.
Step 5.2.2.1.2
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
Add to both sides of the equation.
Step 5.3.2
Add and .
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
Multiply by .
Step 6.2.1.2
Divide by .
Step 6.2.1.3
Subtract from .
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
Simplify each term.
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Step 6.4.1.1.1
Cancel the common factor of and .
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Step 6.4.1.1.1.1
Factor out of .
Step 6.4.1.1.1.2
Cancel the common factors.
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Step 6.4.1.1.1.2.1
Factor out of .
Step 6.4.1.1.1.2.2
Cancel the common factor.
Step 6.4.1.1.1.2.3
Rewrite the expression.
Step 6.4.1.1.1.2.4
Divide by .
Step 6.4.1.1.2
Multiply .
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Step 6.4.1.1.2.1
Multiply by .
Step 6.4.1.1.2.2
Multiply by .
Step 6.4.1.2
Add and .
Step 7
List all of the solutions.