Finite Math Examples

Solve the Matrix Equation [[3a-2c,3b-2d],[2a-c,2b-d]]=[[-2,2],[-1,2]]
Step 1
Write as a linear system of equations.
Step 2
Solve the system of equations.
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Step 2.1
Solve for in .
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Step 2.1.1
Add to both sides of the equation.
Step 2.1.2
Divide each term in by and simplify.
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Step 2.1.2.1
Divide each term in by .
Step 2.1.2.2
Simplify the left side.
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Step 2.1.2.2.1
Cancel the common factor of .
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Step 2.1.2.2.1.1
Cancel the common factor.
Step 2.1.2.2.1.2
Divide by .
Step 2.1.2.3
Simplify the right side.
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Step 2.1.2.3.1
Move the negative in front of the fraction.
Step 2.2
Replace all occurrences of with in each equation.
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Step 2.2.1
Replace all occurrences of in with .
Step 2.2.2
Simplify the left side.
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Step 2.2.2.1
Simplify .
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Step 2.2.2.1.1
Simplify each term.
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Step 2.2.2.1.1.1
Apply the distributive property.
Step 2.2.2.1.1.2
Multiply .
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Step 2.2.2.1.1.2.1
Multiply by .
Step 2.2.2.1.1.2.2
Combine and .
Step 2.2.2.1.1.2.3
Multiply by .
Step 2.2.2.1.1.3
Multiply .
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Step 2.2.2.1.1.3.1
Combine and .
Step 2.2.2.1.1.3.2
Multiply by .
Step 2.2.2.1.1.4
Move the negative in front of the fraction.
Step 2.2.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 2.2.2.1.3
Combine and .
Step 2.2.2.1.4
Combine the numerators over the common denominator.
Step 2.2.2.1.5
Combine the numerators over the common denominator.
Step 2.2.2.1.6
Multiply by .
Step 2.2.2.1.7
Subtract from .
Step 2.2.2.1.8
Rewrite as .
Step 2.2.2.1.9
Factor out of .
Step 2.2.2.1.10
Factor out of .
Step 2.2.2.1.11
Move the negative in front of the fraction.
Step 2.3
Solve for in .
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Step 2.3.1
Multiply both sides of the equation by .
Step 2.3.2
Simplify both sides of the equation.
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Step 2.3.2.1
Simplify the left side.
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Step 2.3.2.1.1
Simplify .
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Step 2.3.2.1.1.1
Cancel the common factor of .
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Step 2.3.2.1.1.1.1
Move the leading negative in into the numerator.
Step 2.3.2.1.1.1.2
Factor out of .
Step 2.3.2.1.1.1.3
Cancel the common factor.
Step 2.3.2.1.1.1.4
Rewrite the expression.
Step 2.3.2.1.1.2
Multiply.
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Step 2.3.2.1.1.2.1
Multiply by .
Step 2.3.2.1.1.2.2
Multiply by .
Step 2.3.2.2
Simplify the right side.
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Step 2.3.2.2.1
Multiply by .
Step 2.3.3
Move all terms not containing to the right side of the equation.
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Step 2.3.3.1
Subtract from both sides of the equation.
Step 2.3.3.2
Subtract from .
Step 2.3.4
Divide each term in by and simplify.
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Step 2.3.4.1
Divide each term in by .
Step 2.3.4.2
Simplify the left side.
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Step 2.3.4.2.1
Dividing two negative values results in a positive value.
Step 2.3.4.2.2
Divide by .
Step 2.3.4.3
Simplify the right side.
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Step 2.3.4.3.1
Divide by .
Step 2.4
Replace all occurrences of with in each equation.
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Step 2.4.1
Replace all occurrences of in with .
Step 2.4.2
Simplify the right side.
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Step 2.4.2.1
Simplify .
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Step 2.4.2.1.1
Combine the numerators over the common denominator.
Step 2.4.2.1.2
Simplify the expression.
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Step 2.4.2.1.2.1
Multiply by .
Step 2.4.2.1.2.2
Add and .
Step 2.4.2.1.2.3
Divide by .
Step 2.5
Solve for in .
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Step 2.5.1
Add to both sides of the equation.
Step 2.5.2
Divide each term in by and simplify.
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Step 2.5.2.1
Divide each term in by .
Step 2.5.2.2
Simplify the left side.
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Step 2.5.2.2.1
Cancel the common factor of .
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Step 2.5.2.2.1.1
Cancel the common factor.
Step 2.5.2.2.1.2
Divide by .
Step 2.6
Replace all occurrences of with in each equation.
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Step 2.6.1
Replace all occurrences of in with .
Step 2.6.2
Simplify the left side.
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Step 2.6.2.1
Simplify .
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Step 2.6.2.1.1
Simplify each term.
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Step 2.6.2.1.1.1
Apply the distributive property.
Step 2.6.2.1.1.2
Multiply .
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Step 2.6.2.1.1.2.1
Combine and .
Step 2.6.2.1.1.2.2
Multiply by .
Step 2.6.2.1.1.3
Multiply .
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Step 2.6.2.1.1.3.1
Combine and .
Step 2.6.2.1.1.3.2
Multiply by .
Step 2.6.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 2.6.2.1.3
Combine and .
Step 2.6.2.1.4
Combine the numerators over the common denominator.
Step 2.6.2.1.5
Combine the numerators over the common denominator.
Step 2.6.2.1.6
Multiply by .
Step 2.6.2.1.7
Subtract from .
Step 2.7
Solve for in .
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Step 2.7.1
Multiply both sides of the equation by .
Step 2.7.2
Simplify both sides of the equation.
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Step 2.7.2.1
Simplify the left side.
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Step 2.7.2.1.1
Cancel the common factor of .
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Step 2.7.2.1.1.1
Cancel the common factor.
Step 2.7.2.1.1.2
Rewrite the expression.
Step 2.7.2.2
Simplify the right side.
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Step 2.7.2.2.1
Multiply by .
Step 2.7.3
Move all terms not containing to the right side of the equation.
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Step 2.7.3.1
Subtract from both sides of the equation.
Step 2.7.3.2
Subtract from .
Step 2.8
Replace all occurrences of with in each equation.
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Step 2.8.1
Replace all occurrences of in with .
Step 2.8.2
Simplify the right side.
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Step 2.8.2.1
Simplify .
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Step 2.8.2.1.1
Combine the numerators over the common denominator.
Step 2.8.2.1.2
Simplify the expression.
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Step 2.8.2.1.2.1
Multiply by .
Step 2.8.2.1.2.2
Add and .
Step 2.8.2.1.2.3
Divide by .
Step 2.9
List all of the solutions.