Linear Algebra Examples

$[\begin{array}{c} 0 & - 1 & 1 \\ 2 & 2 & - 5 \\ - 1 & - 1 & 3 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 8 \\ 15 \\ 23 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 0 & - 1 & 1 \\ 2 & 2 & - 5 \\ - 1 & - 1 & 3 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}]$ .

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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 $3 \times 3$ and the second matrix is $3 \times 1$ .

Step 1.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 0 x - y + 1 z \\ 2 x + 2 y - 5 z \\ - x - y + 3 z \end{array}] = [\begin{array}{c} 8 \\ 15 \\ 23 \end{array}]$

Step 1.3

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} - y + z \\ 2 x + 2 y - 5 z \\ - x - y + 3 z \end{array}] = [\begin{array}{c} 8 \\ 15 \\ 23 \end{array}]$

Step 2

The matrix equation can be written as a set of equations.

$- y + z = 8$

$2 x + 2 y - 5 z = 15$

$- x - y + 3 z = 23$

Step 3

Add $y$ to both sides of the equation.

$z = 8 + y$

$2 x + 2 y - 5 z = 15$

$- x - y + 3 z = 23$

Step 4

Replace all occurrences of $z$ with $8 + y$ in each equation.

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Step 4.1

Replace all occurrences of $z$ in $2 x + 2 y - 5 z = 15$ with $8 + y$ .

$2 x + 2 y - 5 (8 + y) = 15$

$z = 8 + y$

$- x - y + 3 z = 23$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $2 x + 2 y - 5 (8 + y)$ .

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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.

$2 x + 2 y - 5 \cdot 8 - 5 y = 15$

$z = 8 + y$

$- x - y + 3 z = 23$

Step 4.2.1.1.2

Multiply $- 5$ by $8$ .

$2 x + 2 y - 40 - 5 y = 15$

$z = 8 + y$

$- x - y + 3 z = 23$

$2 x + 2 y - 40 - 5 y = 15$

$z = 8 + y$

$- x - y + 3 z = 23$

Step 4.2.1.2

Subtract $5 y$ from $2 y$ .

$2 x - 3 y - 40 = 15$

$z = 8 + y$

$- x - y + 3 z = 23$

$2 x - 3 y - 40 = 15$

$z = 8 + y$

$- x - y + 3 z = 23$

$2 x - 3 y - 40 = 15$

$z = 8 + y$

$- x - y + 3 z = 23$

Step 4.3

Replace all occurrences of $z$ in $- x - y + 3 z = 23$ with $8 + y$ .

$- x - y + 3 (8 + y) = 23$

$2 x - 3 y - 40 = 15$

$z = 8 + y$

Step 4.4

Simplify the left side.

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Step 4.4.1

Simplify $- x - y + 3 (8 + y)$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$- x - y + 3 \cdot 8 + 3 y = 23$

$2 x - 3 y - 40 = 15$

$z = 8 + y$

Step 4.4.1.1.2

Multiply $3$ by $8$ .

$- x - y + 24 + 3 y = 23$

$2 x - 3 y - 40 = 15$

$z = 8 + y$

$- x - y + 24 + 3 y = 23$

$2 x - 3 y - 40 = 15$

$z = 8 + y$

Step 4.4.1.2

Add $- y$ and $3 y$ .

$- x + 2 y + 24 = 23$

$2 x - 3 y - 40 = 15$

$z = 8 + y$

$- x + 2 y + 24 = 23$

$2 x - 3 y - 40 = 15$

$z = 8 + y$

$- x + 2 y + 24 = 23$

$2 x - 3 y - 40 = 15$

$z = 8 + y$

$- x + 2 y + 24 = 23$

$2 x - 3 y - 40 = 15$

$z = 8 + y$

Step 5

Reorder $8$ and $y$ .

$z = y + 8$

$- x + 2 y + 24 = 23$

$2 x - 3 y - 40 = 15$

Step 6

Solve for $y$ in $z = y + 8$ .

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Step 6.1

Rewrite the equation as $y + 8 = z$ .

$y + 8 = z$

$- x + 2 y + 24 = 23$

$2 x - 3 y - 40 = 15$

Step 6.2

Subtract $8$ from both sides of the equation.

$y = z - 8$

$- x + 2 y + 24 = 23$

$2 x - 3 y - 40 = 15$

$y = z - 8$

$- x + 2 y + 24 = 23$

$2 x - 3 y - 40 = 15$

Step 7

Replace all occurrences of $y$ with $z - 8$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $- x + 2 y + 24 = 23$ with $z - 8$ .

$- x + 2 (z - 8) + 24 = 23$

$y = z - 8$

$2 x - 3 y - 40 = 15$

Step 7.2

Simplify the left side.

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Step 7.2.1

Simplify $- x + 2 (z - 8) + 24$ .

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Step 7.2.1.1

Simplify each term.

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Step 7.2.1.1.1

Apply the distributive property.

$- x + 2 z + 2 \cdot - 8 + 24 = 23$

$y = z - 8$

$2 x - 3 y - 40 = 15$

Step 7.2.1.1.2

Multiply $2$ by $- 8$ .

$- x + 2 z - 16 + 24 = 23$

$y = z - 8$

$2 x - 3 y - 40 = 15$

$- x + 2 z - 16 + 24 = 23$

$y = z - 8$

$2 x - 3 y - 40 = 15$

Step 7.2.1.2

Add $- 16$ and $24$ .

$- x + 2 z + 8 = 23$

$y = z - 8$

$2 x - 3 y - 40 = 15$

$- x + 2 z + 8 = 23$

$y = z - 8$

$2 x - 3 y - 40 = 15$

$- x + 2 z + 8 = 23$

$y = z - 8$

$2 x - 3 y - 40 = 15$

Step 7.3

Replace all occurrences of $y$ in $2 x - 3 y - 40 = 15$ with $z - 8$ .

$2 x - 3 (z - 8) - 40 = 15$

$- x + 2 z + 8 = 23$

$y = z - 8$

Step 7.4

Simplify the left side.

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Step 7.4.1

Simplify $2 x - 3 (z - 8) - 40$ .

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Step 7.4.1.1

Simplify each term.

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Step 7.4.1.1.1

Apply the distributive property.

$2 x - 3 z - 3 \cdot - 8 - 40 = 15$

$- x + 2 z + 8 = 23$

$y = z - 8$

Step 7.4.1.1.2

Multiply $- 3$ by $- 8$ .

$2 x - 3 z + 24 - 40 = 15$

$- x + 2 z + 8 = 23$

$y = z - 8$

$2 x - 3 z + 24 - 40 = 15$

$- x + 2 z + 8 = 23$

$y = z - 8$

Step 7.4.1.2

Subtract $40$ from $24$ .

$2 x - 3 z - 16 = 15$

$- x + 2 z + 8 = 23$

$y = z - 8$

$2 x - 3 z - 16 = 15$

$- x + 2 z + 8 = 23$

$y = z - 8$

$2 x - 3 z - 16 = 15$

$- x + 2 z + 8 = 23$

$y = z - 8$

$2 x - 3 z - 16 = 15$

$- x + 2 z + 8 = 23$

$y = z - 8$

Step 8

Solve for $x$ in $2 x - 3 z - 16 = 15$ .

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Step 8.1

Move all terms not containing $x$ to the right side of the equation.

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Step 8.1.1

Add $3 z$ to both sides of the equation.

$2 x - 16 = 15 + 3 z$

$- x + 2 z + 8 = 23$

$y = z - 8$

Step 8.1.2

Add $16$ to both sides of the equation.

$2 x = 15 + 3 z + 16$

$- x + 2 z + 8 = 23$

$y = z - 8$

Step 8.1.3

Add $15$ and $16$ .

$2 x = 3 z + 31$

$- x + 2 z + 8 = 23$

$y = z - 8$

$2 x = 3 z + 31$

$- x + 2 z + 8 = 23$

$y = z - 8$

Step 8.2

Divide each term in $2 x = 3 z + 31$ by $2$ and simplify.

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Step 8.2.1

Divide each term in $2 x = 3 z + 31$ by $2$ .

$\frac{2 x}{2} = \frac{3 z}{2} + \frac{31}{2}$

$- x + 2 z + 8 = 23$

$y = z - 8$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $2$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3 z}{2} + \frac{31}{2}$

$- x + 2 z + 8 = 23$

$y = z - 8$

Step 8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3 z}{2} + \frac{31}{2}$

$- x + 2 z + 8 = 23$

$y = z - 8$

$x = \frac{3 z}{2} + \frac{31}{2}$

$- x + 2 z + 8 = 23$

$y = z - 8$

$x = \frac{3 z}{2} + \frac{31}{2}$

$- x + 2 z + 8 = 23$

$y = z - 8$

$x = \frac{3 z}{2} + \frac{31}{2}$

$- x + 2 z + 8 = 23$

$y = z - 8$

$x = \frac{3 z}{2} + \frac{31}{2}$

$- x + 2 z + 8 = 23$

$y = z - 8$

Step 9

Replace all occurrences of $x$ with $\frac{3 z}{2} + \frac{31}{2}$ in each equation.

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Step 9.1

Replace all occurrences of $x$ in $- x + 2 z + 8 = 23$ with $\frac{3 z}{2} + \frac{31}{2}$ .

$- (\frac{3 z}{2} + \frac{31}{2}) + 2 z + 8 = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 9.2

Simplify the left side.

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Step 9.2.1

Simplify $- (\frac{3 z}{2} + \frac{31}{2}) + 2 z + 8$ .

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Step 9.2.1.1

Apply the distributive property.

$- \frac{3 z}{2} - \frac{31}{2} + 2 z + 8 = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 9.2.1.2

To write $2 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3 z}{2} + 2 z \cdot \frac{2}{2} - \frac{31}{2} + 8 = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 9.2.1.3

Combine $2 z$ and $\frac{2}{2}$ .

$- \frac{3 z}{2} + \frac{2 z \cdot 2}{2} - \frac{31}{2} + 8 = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 9.2.1.4

Combine the numerators over the common denominator.

$\frac{- 3 z + 2 z \cdot 2}{2} - \frac{31}{2} + 8 = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 9.2.1.5

Combine the numerators over the common denominator.

$8 + \frac{- 3 z + 2 z \cdot 2 - 31}{2} = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 9.2.1.6

Multiply $2$ by $2$ .

$8 + \frac{- 3 z + 4 z - 31}{2} = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 9.2.1.7

Add $- 3 z$ and $4 z$ .

$8 + \frac{z - 31}{2} = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 9.2.1.8

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$8 \cdot \frac{2}{2} + \frac{z - 31}{2} = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 9.2.1.9

Simplify terms.

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Step 9.2.1.9.1

Combine $8$ and $\frac{2}{2}$ .

$\frac{8 \cdot 2}{2} + \frac{z - 31}{2} = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 9.2.1.9.2

Combine the numerators over the common denominator.

$\frac{8 \cdot 2 + z - 31}{2} = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

$\frac{8 \cdot 2 + z - 31}{2} = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 9.2.1.10

Simplify the numerator.

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Step 9.2.1.10.1

Multiply $8$ by $2$ .

$\frac{16 + z - 31}{2} = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 9.2.1.10.2

Subtract $31$ from $16$ .

$\frac{z - 15}{2} = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

$\frac{z - 15}{2} = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

$\frac{z - 15}{2} = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

$\frac{z - 15}{2} = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

$\frac{z - 15}{2} = 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 10

Solve for $z$ in $\frac{z - 15}{2} = 23$ .

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Step 10.1

Multiply both sides of the equation by $2$ .

$2 (\frac{z - 15}{2}) = 2 \cdot 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 10.2

Simplify both sides of the equation.

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Step 10.2.1

Simplify the left side.

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Step 10.2.1.1

Cancel the common factor of $2$ .

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Step 10.2.1.1.1

Cancel the common factor.

$2 (\frac{z - 15}{2}) = 2 \cdot 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 10.2.1.1.2

Rewrite the expression.

$z - 15 = 2 \cdot 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

$z - 15 = 2 \cdot 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

$z - 15 = 2 \cdot 23$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 10.2.2

Simplify the right side.

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Step 10.2.2.1

Multiply $2$ by $23$ .

$z - 15 = 46$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

$z - 15 = 46$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

$z - 15 = 46$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 10.3

Move all terms not containing $z$ to the right side of the equation.

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Step 10.3.1

Add $15$ to both sides of the equation.

$z = 46 + 15$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 10.3.2

Add $46$ and $15$ .

$z = 61$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

$z = 61$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

$z = 61$

$x = \frac{3 z}{2} + \frac{31}{2}$

$y = z - 8$

Step 11

Replace all occurrences of $z$ with $61$ in each equation.

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Step 11.1

Replace all occurrences of $z$ in $x = \frac{3 z}{2} + \frac{31}{2}$ with $61$ .

$x = \frac{3 (61)}{2} + \frac{31}{2}$

$z = 61$

$y = z - 8$

Step 11.2

Simplify the right side.

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Step 11.2.1

Simplify $\frac{3 (61)}{2} + \frac{31}{2}$ .

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Step 11.2.1.1

Combine the numerators over the common denominator.

$x = \frac{3 (61) + 31}{2}$

$z = 61$

$y = z - 8$

Step 11.2.1.2

Simplify the expression.

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Step 11.2.1.2.1

Multiply $3$ by $61$ .

$x = \frac{183 + 31}{2}$

$z = 61$

$y = z - 8$

Step 11.2.1.2.2

Add $183$ and $31$ .

$x = \frac{214}{2}$

$z = 61$

$y = z - 8$

Step 11.2.1.2.3

Divide $214$ by $2$ .

$x = 107$

$z = 61$

$y = z - 8$

$x = 107$

$z = 61$

$y = z - 8$

$x = 107$

$z = 61$

$y = z - 8$

$x = 107$

$z = 61$

$y = z - 8$

Step 11.3

Replace all occurrences of $z$ in $y = z - 8$ with $61$ .

$y = (61) - 8$

$x = 107$

$z = 61$

Step 11.4

Simplify the right side.

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Step 11.4.1

Subtract $8$ from $61$ .

$y = 53$

$x = 107$

$z = 61$

$y = 53$

$x = 107$

$z = 61$

$y = 53$

$x = 107$

$z = 61$

Step 12

List all of the solutions.

$y = 53, x = 107, z = 61$