Linear Algebra Examples

Solve the Matrix Equation [[-1/2,3/10,3/5],[0,-2/5,1/5],[1/2,-1/10,-1/5]][[x],[y],[z]]=[[19],[6],[8]]
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.
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Step 1.3.1
Combine and .
Step 1.3.2
Combine and .
Step 1.3.3
Combine and .
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
Move all terms not containing to the right side of the equation.
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Step 3.1.1.1
Subtract from both sides of the equation.
Step 3.1.1.2
Subtract from both sides of the equation.
Step 3.1.2
Multiply both sides of the equation by .
Step 3.1.3
Simplify both sides of the equation.
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Step 3.1.3.1
Simplify the left side.
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Step 3.1.3.1.1
Simplify .
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Step 3.1.3.1.1.1
Cancel the common factor of .
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Step 3.1.3.1.1.1.1
Move the leading negative in into the numerator.
Step 3.1.3.1.1.1.2
Factor out of .
Step 3.1.3.1.1.1.3
Cancel the common factor.
Step 3.1.3.1.1.1.4
Rewrite the expression.
Step 3.1.3.1.1.2
Multiply.
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Step 3.1.3.1.1.2.1
Multiply by .
Step 3.1.3.1.1.2.2
Multiply by .
Step 3.1.3.2
Simplify the right side.
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Step 3.1.3.2.1
Simplify .
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Step 3.1.3.2.1.1
Apply the distributive property.
Step 3.1.3.2.1.2
Simplify.
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Step 3.1.3.2.1.2.1
Multiply by .
Step 3.1.3.2.1.2.2
Cancel the common factor of .
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Step 3.1.3.2.1.2.2.1
Move the leading negative in into the numerator.
Step 3.1.3.2.1.2.2.2
Factor out of .
Step 3.1.3.2.1.2.2.3
Factor out of .
Step 3.1.3.2.1.2.2.4
Cancel the common factor.
Step 3.1.3.2.1.2.2.5
Rewrite the expression.
Step 3.1.3.2.1.2.3
Multiply .
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Step 3.1.3.2.1.2.3.1
Multiply by .
Step 3.1.3.2.1.2.3.2
Combine and .
Step 3.1.3.2.1.2.3.3
Multiply by .
Step 3.1.3.2.1.3
Simplify each term.
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Step 3.1.3.2.1.3.1
Move the negative in front of the fraction.
Step 3.1.3.2.1.3.2
Multiply .
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Step 3.1.3.2.1.3.2.1
Multiply by .
Step 3.1.3.2.1.3.2.2
Multiply by .
Step 3.1.4
Move .
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
Simplify the numerator.
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Step 3.2.2.1.1.1.1
Combine the numerators over the common denominator.
Step 3.2.2.1.1.1.2
Factor out of .
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Step 3.2.2.1.1.1.2.1
Factor out of .
Step 3.2.2.1.1.1.2.2
Factor out of .
Step 3.2.2.1.1.1.2.3
Factor out of .
Step 3.2.2.1.1.1.3
To write as a fraction with a common denominator, multiply by .
Step 3.2.2.1.1.1.4
Combine and .
Step 3.2.2.1.1.1.5
Combine the numerators over the common denominator.
Step 3.2.2.1.1.1.6
Simplify the numerator.
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Step 3.2.2.1.1.1.6.1
Apply the distributive property.
Step 3.2.2.1.1.1.6.2
Multiply by .
Step 3.2.2.1.1.1.6.3
Multiply by .
Step 3.2.2.1.1.2
Multiply the numerator by the reciprocal of the denominator.
Step 3.2.2.1.1.3
Multiply .
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Step 3.2.2.1.1.3.1
Multiply by .
Step 3.2.2.1.1.3.2
Multiply by .
Step 3.2.2.1.2
Simplify terms.
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Step 3.2.2.1.2.1
Combine the numerators over the common denominator.
Step 3.2.2.1.2.2
Subtract from .
Step 3.2.2.1.3
Simplify each term.
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Step 3.2.2.1.3.1
Cancel the common factor of and .
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Step 3.2.2.1.3.1.1
Factor out of .
Step 3.2.2.1.3.1.2
Factor out of .
Step 3.2.2.1.3.1.3
Factor out of .
Step 3.2.2.1.3.1.4
Factor out of .
Step 3.2.2.1.3.1.5
Factor out of .
Step 3.2.2.1.3.1.6
Cancel the common factors.
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Step 3.2.2.1.3.1.6.1
Factor out of .
Step 3.2.2.1.3.1.6.2
Cancel the common factor.
Step 3.2.2.1.3.1.6.3
Rewrite the expression.
Step 3.2.2.1.3.2
Move the negative in front of the fraction.
Step 3.2.2.1.4
Simplify terms.
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Step 3.2.2.1.4.1
Combine the numerators over the common denominator.
Step 3.2.2.1.4.2
Subtract from .
Step 3.3
Solve for in .
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Step 3.3.1
Multiply both sides by .
Step 3.3.2
Simplify.
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Step 3.3.2.1
Simplify the left side.
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Step 3.3.2.1.1
Cancel the common factor of .
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Step 3.3.2.1.1.1
Cancel the common factor.
Step 3.3.2.1.1.2
Rewrite the expression.
Step 3.3.2.2
Simplify the right side.
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Step 3.3.2.2.1
Multiply by .
Step 3.3.3
Move all terms not containing to the right side of the equation.
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Step 3.3.3.1
Subtract from both sides of the equation.
Step 3.3.3.2
Add to both sides of the equation.
Step 3.3.3.3
Add and .
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
Combine the numerators over the common denominator.
Step 3.4.2.1.2
Simplify each term.
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Step 3.4.2.1.2.1
Apply the distributive property.
Step 3.4.2.1.2.2
Multiply by .
Step 3.4.2.1.2.3
Multiply by .
Step 3.4.2.1.3
Simplify by adding terms.
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Step 3.4.2.1.3.1
Combine the opposite terms in .
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Step 3.4.2.1.3.1.1
Add and .
Step 3.4.2.1.3.1.2
Add and .
Step 3.4.2.1.3.2
Simplify the expression.
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Step 3.4.2.1.3.2.1
Divide by .
Step 3.4.2.1.3.2.2
Add and .
Step 3.4.3
Replace all occurrences of in with .
Step 3.4.4
Simplify the left side.
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Step 3.4.4.1
Simplify .
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Step 3.4.4.1.1
Combine the numerators over the common denominator.
Step 3.4.4.1.2
Simplify each term.
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Step 3.4.4.1.2.1
Apply the distributive property.
Step 3.4.4.1.2.2
Multiply by .
Step 3.4.4.1.2.3
Multiply by .
Step 3.4.4.1.3
Simplify terms.
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Step 3.4.4.1.3.1
Add and .
Step 3.4.4.1.3.2
Cancel the common factor of and .
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Step 3.4.4.1.3.2.1
Factor out of .
Step 3.4.4.1.3.2.2
Factor out of .
Step 3.4.4.1.3.2.3
Factor out of .
Step 3.4.4.1.3.2.4
Cancel the common factors.
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Step 3.4.4.1.3.2.4.1
Factor out of .
Step 3.4.4.1.3.2.4.2
Cancel the common factor.
Step 3.4.4.1.3.2.4.3
Rewrite the expression.
Step 3.4.4.1.3.2.4.4
Divide by .
Step 3.5
Move all terms not containing to the right side of the equation.
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Step 3.5.1
Add to both sides of the equation.
Step 3.5.2
Add and .
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 right side.
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Step 3.6.2.1
Simplify .
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Step 3.6.2.1.1
Multiply by .
Step 3.6.2.1.2
Add and .
Step 3.7
List all of the solutions.