Finite Math Examples

Solve the Matrix Equation [[4,-1],[3,-3]]*[[a,b],[c,d]]=[[8,15],[-12,9]]
Step 1
Multiply .
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Step 1.1
Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is and the second matrix is .
Step 1.2
Multiply each row in the first matrix by each column in the second matrix.
Step 2
Write as a linear system of equations.
Step 3
Solve the system of equations.
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Step 3.1
Solve for in .
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Step 3.1.1
Add to both sides of the equation.
Step 3.1.2
Divide each term in by and simplify.
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Step 3.1.2.1
Divide each term in by .
Step 3.1.2.2
Simplify the left side.
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Step 3.1.2.2.1
Cancel the common factor of .
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Step 3.1.2.2.1.1
Cancel the common factor.
Step 3.1.2.2.1.2
Divide by .
Step 3.1.2.3
Simplify the right side.
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Step 3.1.2.3.1
Divide by .
Step 3.2
Replace all occurrences of with in each equation.
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Step 3.2.1
Replace all occurrences of in with .
Step 3.2.2
Simplify the left side.
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Step 3.2.2.1
Simplify .
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Step 3.2.2.1.1
Simplify each term.
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Step 3.2.2.1.1.1
Apply the distributive property.
Step 3.2.2.1.1.2
Multiply by .
Step 3.2.2.1.1.3
Combine and .
Step 3.2.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 3.2.2.1.3
Simplify terms.
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Step 3.2.2.1.3.1
Combine and .
Step 3.2.2.1.3.2
Combine the numerators over the common denominator.
Step 3.2.2.1.4
Simplify each term.
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Step 3.2.2.1.4.1
Simplify the numerator.
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Step 3.2.2.1.4.1.1
Factor out of .
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Step 3.2.2.1.4.1.1.1
Factor out of .
Step 3.2.2.1.4.1.1.2
Factor out of .
Step 3.2.2.1.4.1.1.3
Factor out of .
Step 3.2.2.1.4.1.2
Multiply by .
Step 3.2.2.1.4.1.3
Subtract from .
Step 3.2.2.1.4.1.4
Multiply by .
Step 3.2.2.1.4.2
Move the negative in front of the fraction.
Step 3.3
Solve for in .
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Step 3.3.1
Move all terms not containing to the right side of the equation.
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Step 3.3.1.1
Subtract from both sides of the equation.
Step 3.3.1.2
Subtract from .
Step 3.3.2
Multiply both sides of the equation by .
Step 3.3.3
Simplify both sides of the equation.
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Step 3.3.3.1
Simplify the left side.
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Step 3.3.3.1.1
Simplify .
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Step 3.3.3.1.1.1
Cancel the common factor of .
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Step 3.3.3.1.1.1.1
Move the leading negative in into the numerator.
Step 3.3.3.1.1.1.2
Move the leading negative in into the numerator.
Step 3.3.3.1.1.1.3
Factor out of .
Step 3.3.3.1.1.1.4
Cancel the common factor.
Step 3.3.3.1.1.1.5
Rewrite the expression.
Step 3.3.3.1.1.2
Cancel the common factor of .
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Step 3.3.3.1.1.2.1
Factor out of .
Step 3.3.3.1.1.2.2
Cancel the common factor.
Step 3.3.3.1.1.2.3
Rewrite the expression.
Step 3.3.3.1.1.3
Multiply.
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Step 3.3.3.1.1.3.1
Multiply by .
Step 3.3.3.1.1.3.2
Multiply by .
Step 3.3.3.2
Simplify the right side.
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Step 3.3.3.2.1
Simplify .
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Step 3.3.3.2.1.1
Cancel the common factor of .
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Step 3.3.3.2.1.1.1
Move the leading negative in into the numerator.
Step 3.3.3.2.1.1.2
Factor out of .
Step 3.3.3.2.1.1.3
Cancel the common factor.
Step 3.3.3.2.1.1.4
Rewrite the expression.
Step 3.3.3.2.1.2
Multiply by .
Step 3.4
Replace all occurrences of with in each equation.
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Step 3.4.1
Replace all occurrences of in with .
Step 3.4.2
Simplify the right side.
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Step 3.4.2.1
Simplify .
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Step 3.4.2.1.1
Divide by .
Step 3.4.2.1.2
Add and .
Step 3.5
Solve for in .
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Step 3.5.1
Add to both sides of the equation.
Step 3.5.2
Divide each term in by and simplify.
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Step 3.5.2.1
Divide each term in by .
Step 3.5.2.2
Simplify the left side.
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Step 3.5.2.2.1
Cancel the common factor of .
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Step 3.5.2.2.1.1
Cancel the common factor.
Step 3.5.2.2.1.2
Divide by .
Step 3.6
Replace all occurrences of with in each equation.
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Step 3.6.1
Replace all occurrences of in with .
Step 3.6.2
Simplify the left side.
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Step 3.6.2.1
Simplify .
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Step 3.6.2.1.1
Simplify each term.
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Step 3.6.2.1.1.1
Apply the distributive property.
Step 3.6.2.1.1.2
Multiply .
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Step 3.6.2.1.1.2.1
Combine and .
Step 3.6.2.1.1.2.2
Multiply by .
Step 3.6.2.1.1.3
Combine and .
Step 3.6.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 3.6.2.1.3
Combine and .
Step 3.6.2.1.4
Combine the numerators over the common denominator.
Step 3.6.2.1.5
Combine the numerators over the common denominator.
Step 3.6.2.1.6
Multiply by .
Step 3.6.2.1.7
Subtract from .
Step 3.6.2.1.8
Factor out of .
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Step 3.6.2.1.8.1
Factor out of .
Step 3.6.2.1.8.2
Factor out of .
Step 3.6.2.1.8.3
Factor out of .
Step 3.7
Solve for in .
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Step 3.7.1
Multiply both sides by .
Step 3.7.2
Simplify.
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Step 3.7.2.1
Simplify the left side.
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Step 3.7.2.1.1
Simplify .
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Step 3.7.2.1.1.1
Cancel the common factor of .
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Step 3.7.2.1.1.1.1
Cancel the common factor.
Step 3.7.2.1.1.1.2
Rewrite the expression.
Step 3.7.2.1.1.2
Apply the distributive property.
Step 3.7.2.1.1.3
Simplify the expression.
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Step 3.7.2.1.1.3.1
Multiply by .
Step 3.7.2.1.1.3.2
Multiply by .
Step 3.7.2.1.1.3.3
Reorder and .
Step 3.7.2.2
Simplify the right side.
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Step 3.7.2.2.1
Multiply by .
Step 3.7.3
Solve for .
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Step 3.7.3.1
Move all terms not containing to the right side of the equation.
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Step 3.7.3.1.1
Subtract from both sides of the equation.
Step 3.7.3.1.2
Subtract from .
Step 3.7.3.2
Divide each term in by and simplify.
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Step 3.7.3.2.1
Divide each term in by .
Step 3.7.3.2.2
Simplify the left side.
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Step 3.7.3.2.2.1
Cancel the common factor of .
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Step 3.7.3.2.2.1.1
Cancel the common factor.
Step 3.7.3.2.2.1.2
Divide by .
Step 3.7.3.2.3
Simplify the right side.
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Step 3.7.3.2.3.1
Divide by .
Step 3.8
Replace all occurrences of with in each equation.
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Step 3.8.1
Replace all occurrences of in with .
Step 3.8.2
Simplify the right side.
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Step 3.8.2.1
Simplify .
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Step 3.8.2.1.1
Combine the numerators over the common denominator.
Step 3.8.2.1.2
Simplify the expression.
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Step 3.8.2.1.2.1
Add and .
Step 3.8.2.1.2.2
Divide by .
Step 3.9
List all of the solutions.