Linear Algebra Examples

$[\begin{array}{c} 4 \\ 8 \\ 6 \end{array}] + 2 [\begin{array}{c} x \\ y \\ 5 z \end{array}] = [\begin{array}{c} - 4 \\ - 22 \\ 16 \end{array}]$

Step 1

Simplify each term.

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Step 1.1

Multiply $2$ by each element of the matrix.

$[\begin{array}{c} 4 \\ 8 \\ 6 \end{array}] + [\begin{array}{c} 2 x \\ 2 y \\ 2 (5 z) \end{array}] = [\begin{array}{c} - 4 \\ - 22 \\ 16 \end{array}]$

Step 1.2

Multiply $5$ by $2$ .

$[\begin{array}{c} 4 \\ 8 \\ 6 \end{array}] + [\begin{array}{c} 2 x \\ 2 y \\ 10 z \end{array}] = [\begin{array}{c} - 4 \\ - 22 \\ 16 \end{array}]$

Step 2

Add the corresponding elements.

$[\begin{array}{c} 4 + 2 x \\ 8 + 2 y \\ 6 + 10 z \end{array}] = [\begin{array}{c} - 4 \\ - 22 \\ 16 \end{array}]$

Step 3

Write as a linear system of equations.

$4 + 2 x = - 4$

$8 + 2 y = - 22$

$6 + 10 z = 16$

Step 4

Solve the system of equations.

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Step 4.1

Solve for $x$ in $4 + 2 x = - 4$ .

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Step 4.1.1

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

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Step 4.1.1.1

Subtract $4$ from both sides of the equation.

$2 x = - 4 - 4$

$8 + 2 y = - 22$

$6 + 10 z = 16$

Step 4.1.1.2

Subtract $4$ from $- 4$ .

$2 x = - 8$

$8 + 2 y = - 22$

$6 + 10 z = 16$

$2 x = - 8$

$8 + 2 y = - 22$

$6 + 10 z = 16$

Step 4.1.2

Divide each term in $2 x = - 8$ by $2$ and simplify.

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Step 4.1.2.1

Divide each term in $2 x = - 8$ by $2$ .

$\frac{2 x}{2} = \frac{- 8}{2}$

$8 + 2 y = - 22$

$6 + 10 z = 16$

Step 4.1.2.2

Simplify the left side.

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Step 4.1.2.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 8}{2}$

$8 + 2 y = - 22$

$6 + 10 z = 16$

Step 4.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{2}$

$8 + 2 y = - 22$

$6 + 10 z = 16$

$x = \frac{- 8}{2}$

$8 + 2 y = - 22$

$6 + 10 z = 16$

$x = \frac{- 8}{2}$

$8 + 2 y = - 22$

$6 + 10 z = 16$

Step 4.1.2.3

Simplify the right side.

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Step 4.1.2.3.1

Divide $- 8$ by $2$ .

$x = - 4$

$8 + 2 y = - 22$

$6 + 10 z = 16$

$x = - 4$

$8 + 2 y = - 22$

$6 + 10 z = 16$

$x = - 4$

$8 + 2 y = - 22$

$6 + 10 z = 16$

$x = - 4$

$8 + 2 y = - 22$

$6 + 10 z = 16$

Step 4.2

Replace all occurrences of $x$ with $- 4$ in each equation.

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Step 4.2.1

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

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Step 4.2.1.1

Subtract $8$ from both sides of the equation.

$2 y = - 22 - 8$

$x = - 4$

$6 + 10 z = 16$

Step 4.2.1.2

Subtract $8$ from $- 22$ .

$2 y = - 30$

$x = - 4$

$6 + 10 z = 16$

$2 y = - 30$

$x = - 4$

$6 + 10 z = 16$

Step 4.2.2

Divide each term in $2 y = - 30$ by $2$ and simplify.

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Step 4.2.2.1

Divide each term in $2 y = - 30$ by $2$ .

$\frac{2 y}{2} = \frac{- 30}{2}$

$x = - 4$

$6 + 10 z = 16$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 30}{2}$

$x = - 4$

$6 + 10 z = 16$

Step 4.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 30}{2}$

$x = - 4$

$6 + 10 z = 16$

$y = \frac{- 30}{2}$

$x = - 4$

$6 + 10 z = 16$

$y = \frac{- 30}{2}$

$x = - 4$

$6 + 10 z = 16$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Divide $- 30$ by $2$ .

$y = - 15$

$x = - 4$

$6 + 10 z = 16$

$y = - 15$

$x = - 4$

$6 + 10 z = 16$

$y = - 15$

$x = - 4$

$6 + 10 z = 16$

Step 4.2.3

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

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Step 4.2.3.1

Subtract $6$ from both sides of the equation.

$10 z = 16 - 6$

$y = - 15$

$x = - 4$

Step 4.2.3.2

Subtract $6$ from $16$ .

$10 z = 10$

$y = - 15$

$x = - 4$

$10 z = 10$

$y = - 15$

$x = - 4$

Step 4.2.4

Divide each term in $10 z = 10$ by $10$ and simplify.

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Step 4.2.4.1

Divide each term in $10 z = 10$ by $10$ .

$\frac{10 z}{10} = \frac{10}{10}$

$y = - 15$

$x = - 4$

Step 4.2.4.2

Simplify the left side.

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Step 4.2.4.2.1

Cancel the common factor of $10$ .

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Step 4.2.4.2.1.1

Cancel the common factor.

$\frac{10 z}{10} = \frac{10}{10}$

$y = - 15$

$x = - 4$

Step 4.2.4.2.1.2

Divide $z$ by $1$ .

$z = \frac{10}{10}$

$y = - 15$

$x = - 4$

$z = \frac{10}{10}$

$y = - 15$

$x = - 4$

$z = \frac{10}{10}$

$y = - 15$

$x = - 4$

Step 4.2.4.3

Simplify the right side.

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Step 4.2.4.3.1

Divide $10$ by $10$ .

$z = 1$

$y = - 15$

$x = - 4$

$z = 1$

$y = - 15$

$x = - 4$

$z = 1$

$y = - 15$

$x = - 4$

$z = 1$

$y = - 15$

$x = - 4$

Step 4.3

List all of the solutions.

$z = 1, y = - 15, x = - 4$