Finite Math Examples

Solve the Matrix Equation [[-4,-2],[3,-5]][[x],[y]]+[[13],[-14]]=[[23],[-28]]
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
Add the corresponding elements.
Step 3
Write as a linear system of equations.
Step 4
Solve the system of equations.
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Step 4.1
Solve for in .
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Step 4.1.1
Move all terms not containing to the right side of the equation.
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Step 4.1.1.1
Add to both sides of the equation.
Step 4.1.1.2
Subtract from both sides of the equation.
Step 4.1.1.3
Subtract from .
Step 4.1.2
Divide each term in by and simplify.
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Step 4.1.2.1
Divide each term in by .
Step 4.1.2.2
Simplify the left side.
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Step 4.1.2.2.1
Cancel the common factor of .
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Step 4.1.2.2.1.1
Cancel the common factor.
Step 4.1.2.2.1.2
Divide by .
Step 4.1.2.3
Simplify the right side.
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Step 4.1.2.3.1
Simplify each term.
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Step 4.1.2.3.1.1
Cancel the common factor of and .
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Step 4.1.2.3.1.1.1
Factor out of .
Step 4.1.2.3.1.1.2
Cancel the common factors.
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Step 4.1.2.3.1.1.2.1
Factor out of .
Step 4.1.2.3.1.1.2.2
Cancel the common factor.
Step 4.1.2.3.1.1.2.3
Rewrite the expression.
Step 4.1.2.3.1.2
Move the negative in front of the fraction.
Step 4.1.2.3.1.3
Cancel the common factor of and .
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Step 4.1.2.3.1.3.1
Factor out of .
Step 4.1.2.3.1.3.2
Cancel the common factors.
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Step 4.1.2.3.1.3.2.1
Factor out of .
Step 4.1.2.3.1.3.2.2
Cancel the common factor.
Step 4.1.2.3.1.3.2.3
Rewrite the expression.
Step 4.1.2.3.1.4
Move the negative in front of the fraction.
Step 4.2
Replace all occurrences of with in each equation.
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Step 4.2.1
Replace all occurrences of in with .
Step 4.2.2
Simplify the left side.
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Step 4.2.2.1
Simplify .
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Step 4.2.2.1.1
Simplify each term.
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Step 4.2.2.1.1.1
Apply the distributive property.
Step 4.2.2.1.1.2
Multiply .
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Step 4.2.2.1.1.2.1
Multiply by .
Step 4.2.2.1.1.2.2
Combine and .
Step 4.2.2.1.1.3
Multiply .
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Step 4.2.2.1.1.3.1
Multiply by .
Step 4.2.2.1.1.3.2
Combine and .
Step 4.2.2.1.1.3.3
Multiply by .
Step 4.2.2.1.1.4
Simplify each term.
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Step 4.2.2.1.1.4.1
Move the negative in front of the fraction.
Step 4.2.2.1.1.4.2
Move the negative in front of the fraction.
Step 4.2.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 4.2.2.1.3
Combine and .
Step 4.2.2.1.4
Combine the numerators over the common denominator.
Step 4.2.2.1.5
Combine the numerators over the common denominator.
Step 4.2.2.1.6
Multiply by .
Step 4.2.2.1.7
Subtract from .
Step 4.2.2.1.8
To write as a fraction with a common denominator, multiply by .
Step 4.2.2.1.9
Simplify terms.
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Step 4.2.2.1.9.1
Combine and .
Step 4.2.2.1.9.2
Combine the numerators over the common denominator.
Step 4.2.2.1.10
Simplify the numerator.
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Step 4.2.2.1.10.1
Multiply by .
Step 4.2.2.1.10.2
Subtract from .
Step 4.2.2.1.11
Simplify with factoring out.
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Step 4.2.2.1.11.1
Factor out of .
Step 4.2.2.1.11.2
Rewrite as .
Step 4.2.2.1.11.3
Factor out of .
Step 4.2.2.1.11.4
Simplify the expression.
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Step 4.2.2.1.11.4.1
Rewrite as .
Step 4.2.2.1.11.4.2
Move the negative in front of the fraction.
Step 4.3
Solve for in .
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Step 4.3.1
Multiply both sides of the equation by .
Step 4.3.2
Simplify both sides of the equation.
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Step 4.3.2.1
Simplify the left side.
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Step 4.3.2.1.1
Simplify .
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Step 4.3.2.1.1.1
Cancel the common factor of .
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Step 4.3.2.1.1.1.1
Move the leading negative in into the numerator.
Step 4.3.2.1.1.1.2
Factor out of .
Step 4.3.2.1.1.1.3
Cancel the common factor.
Step 4.3.2.1.1.1.4
Rewrite the expression.
Step 4.3.2.1.1.2
Multiply.
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Step 4.3.2.1.1.2.1
Multiply by .
Step 4.3.2.1.1.2.2
Multiply by .
Step 4.3.2.2
Simplify the right side.
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Step 4.3.2.2.1
Multiply by .
Step 4.3.3
Move all terms not containing to the right side of the equation.
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Step 4.3.3.1
Subtract from both sides of the equation.
Step 4.3.3.2
Subtract from .
Step 4.3.4
Divide each term in by and simplify.
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Step 4.3.4.1
Divide each term in by .
Step 4.3.4.2
Simplify the left side.
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Step 4.3.4.2.1
Cancel the common factor of .
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Step 4.3.4.2.1.1
Cancel the common factor.
Step 4.3.4.2.1.2
Divide by .
Step 4.3.4.3
Simplify the right side.
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Step 4.3.4.3.1
Divide by .
Step 4.4
Replace all occurrences of with in each equation.
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Step 4.4.1
Replace all occurrences of in with .
Step 4.4.2
Simplify the right side.
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Step 4.4.2.1
Simplify .
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Step 4.4.2.1.1
Combine the numerators over the common denominator.
Step 4.4.2.1.2
Simplify the expression.
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Step 4.4.2.1.2.1
Subtract from .
Step 4.4.2.1.2.2
Divide by .
Step 4.5
List all of the solutions.