Linear Algebra Examples

Find the Variables [[-3,1,2],[4,2,-4],[0,1,3]]*[[x],[y],[z]]=[[5],[-4],[3]]
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 1.3
Simplify each element of the matrix by multiplying out all the expressions.
Step 2
The matrix equation can be written as a set of equations.
Step 3
Move all terms not containing to the right side of the equation.
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Step 3.1
Add to both sides of the equation.
Step 3.2
Subtract from both sides of the equation.
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
Simplify.
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Step 4.2.1.1.2.1
Multiply by .
Step 4.2.1.1.2.2
Multiply by .
Step 4.2.1.1.2.3
Multiply by .
Step 4.2.1.2
Simplify by adding terms.
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Step 4.2.1.2.1
Add and .
Step 4.2.1.2.2
Subtract from .
Step 4.3
Replace all occurrences of in with .
Step 4.4
Simplify the left side.
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Step 4.4.1
Add and .
Step 5
Move all terms not containing to the right side of the equation.
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Step 5.1
Subtract from both sides of the equation.
Step 5.2
Subtract from both sides of the equation.
Step 5.3
Subtract from .
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 left side.
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Step 6.2.1
Simplify .
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Step 6.2.1.1
Simplify each term.
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Step 6.2.1.1.1
Apply the distributive property.
Step 6.2.1.1.2
Multiply by .
Step 6.2.1.1.3
Multiply by .
Step 6.2.1.2
Simplify by adding terms.
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Step 6.2.1.2.1
Add and .
Step 6.2.1.2.2
Add and .
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
Apply the distributive property.
Step 6.4.1.1.2
Multiply by .
Step 6.4.1.1.3
Multiply by .
Step 6.4.1.2
Simplify by adding terms.
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Step 6.4.1.2.1
Add and .
Step 6.4.1.2.2
Add and .
Step 7
Solve for in .
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Step 7.1
Move all terms not containing to the right side of the equation.
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Step 7.1.1
Subtract from both sides of the equation.
Step 7.1.2
Subtract from .
Step 7.2
Divide each term in by and simplify.
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Step 7.2.1
Divide each term in by .
Step 7.2.2
Simplify the left side.
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Step 7.2.2.1
Cancel the common factor of .
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Step 7.2.2.1.1
Cancel the common factor.
Step 7.2.2.1.2
Divide by .
Step 7.2.3
Simplify the right side.
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Step 7.2.3.1
Cancel the common factor of and .
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Step 7.2.3.1.1
Factor out of .
Step 7.2.3.1.2
Cancel the common factors.
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Step 7.2.3.1.2.1
Factor out of .
Step 7.2.3.1.2.2
Cancel the common factor.
Step 7.2.3.1.2.3
Rewrite the expression.
Step 7.2.3.2
Move the negative in front of the fraction.
Step 8
Replace all occurrences of with in each equation.
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Step 8.1
Replace all occurrences of in with .
Step 8.2
Simplify the right side.
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Step 8.2.1
Simplify .
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Step 8.2.1.1
Simplify each term.
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Step 8.2.1.1.1
Multiply .
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Step 8.2.1.1.1.1
Multiply by .
Step 8.2.1.1.1.2
Combine and .
Step 8.2.1.1.1.3
Multiply by .
Step 8.2.1.1.2
Move the negative in front of the fraction.
Step 8.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 8.2.1.3
Combine and .
Step 8.2.1.4
Combine the numerators over the common denominator.
Step 8.2.1.5
Simplify the numerator.
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Step 8.2.1.5.1
Multiply by .
Step 8.2.1.5.2
Add and .
Step 8.3
Replace all occurrences of in with .
Step 8.4
Simplify the right side.
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Step 8.4.1
Simplify .
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Step 8.4.1.1
Multiply .
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Step 8.4.1.1.1
Multiply by .
Step 8.4.1.1.2
Combine and .
Step 8.4.1.1.3
Multiply by .
Step 8.4.1.2
To write as a fraction with a common denominator, multiply by .
Step 8.4.1.3
Combine and .
Step 8.4.1.4
Combine the numerators over the common denominator.
Step 8.4.1.5
Simplify the numerator.
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Step 8.4.1.5.1
Multiply by .
Step 8.4.1.5.2
Add and .
Step 9
List all of the solutions.