Linear Algebra Examples

$[\begin{array}{c} 3 & - 2 & 16 \\ 2 & 0 & - 8 \\ 0 & 0 & 1 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 6 \\ 7 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 3 & - 2 & 16 \\ 2 & 0 & - 8 \\ 0 & 0 & 1 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}]$ .

Tap for more steps...

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.

$[\begin{array}{c} 3 x - 2 y + 16 z \\ 2 x + 0 y - 8 z \\ 0 x + 0 y + 1 z \end{array}] = [\begin{array}{c} 0 \\ 6 \\ 7 \end{array}]$

Step 1.3

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

Tap for more steps...

Step 1.3.1

Add $0$ and $0$ .

$[\begin{array}{c} 3 x - 2 y + 16 z \\ 2 x - 8 z \\ 0 + z \end{array}] = [\begin{array}{c} 0 \\ 6 \\ 7 \end{array}]$

Step 1.3.2

Add $0$ and $z$ .

$[\begin{array}{c} 3 x - 2 y + 16 z \\ 2 x - 8 z \\ z \end{array}] = [\begin{array}{c} 0 \\ 6 \\ 7 \end{array}]$

Step 2

Write as a linear system of equations.

$3 x - 2 y + 16 z = 0$

$2 x - 8 z = 6$

$z = 7$

Step 3

Solve the system of equations.

Tap for more steps...

Step 3.1

Replace all occurrences of $z$ with $7$ in each equation.

Tap for more steps...

Step 3.1.1

Replace all occurrences of $z$ in $3 x - 2 y + 16 z = 0$ with $7$ .

$3 x - 2 y + 16 (7) = 0$

$z = 7$

$2 x - 8 z = 6$

Step 3.1.2

Simplify the left side.

Tap for more steps...

Step 3.1.2.1

Multiply $16$ by $7$ .

$3 x - 2 y + 112 = 0$

$z = 7$

$2 x - 8 z = 6$

$3 x - 2 y + 112 = 0$

$z = 7$

$2 x - 8 z = 6$

Step 3.1.3

Replace all occurrences of $z$ in $2 x - 8 z = 6$ with $7$ .

$2 x - 8 \cdot 7 = 6$

$3 x - 2 y + 112 = 0$

$z = 7$

Step 3.1.4

Simplify the left side.

Tap for more steps...

Step 3.1.4.1

Multiply $- 8$ by $7$ .

$2 x - 56 = 6$

$3 x - 2 y + 112 = 0$

$z = 7$

$2 x - 56 = 6$

$3 x - 2 y + 112 = 0$

$z = 7$

$2 x - 56 = 6$

$3 x - 2 y + 112 = 0$

$z = 7$

Step 3.2

Solve for $x$ in $2 x - 56 = 6$ .

Tap for more steps...

Step 3.2.1

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

Tap for more steps...

Step 3.2.1.1

Add $56$ to both sides of the equation.

$2 x = 6 + 56$

$3 x - 2 y + 112 = 0$

$z = 7$

Step 3.2.1.2

Add $6$ and $56$ .

$2 x = 62$

$3 x - 2 y + 112 = 0$

$z = 7$

$2 x = 62$

$3 x - 2 y + 112 = 0$

$z = 7$

Step 3.2.2

Divide each term in $2 x = 62$ by $2$ and simplify.

Tap for more steps...

Step 3.2.2.1

Divide each term in $2 x = 62$ by $2$ .

$\frac{2 x}{2} = \frac{62}{2}$

$3 x - 2 y + 112 = 0$

$z = 7$

Step 3.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{62}{2}$

$3 x - 2 y + 112 = 0$

$z = 7$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{62}{2}$

$3 x - 2 y + 112 = 0$

$z = 7$

$x = \frac{62}{2}$

$3 x - 2 y + 112 = 0$

$z = 7$

$x = \frac{62}{2}$

$3 x - 2 y + 112 = 0$

$z = 7$

Step 3.2.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.2.3.1

Divide $62$ by $2$ .

$x = 31$

$3 x - 2 y + 112 = 0$

$z = 7$

$x = 31$

$3 x - 2 y + 112 = 0$

$z = 7$

$x = 31$

$3 x - 2 y + 112 = 0$

$z = 7$

$x = 31$

$3 x - 2 y + 112 = 0$

$z = 7$

Step 3.3

Replace all occurrences of $x$ with $31$ in each equation.

Tap for more steps...

Step 3.3.1

Replace all occurrences of $x$ in $3 x - 2 y + 112 = 0$ with $31$ .

$3 (31) - 2 y + 112 = 0$

$x = 31$

$z = 7$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Simplify $3 (31) - 2 y + 112$ .

Tap for more steps...

Step 3.3.2.1.1

Multiply $3$ by $31$ .

$93 - 2 y + 112 = 0$

$x = 31$

$z = 7$

Step 3.3.2.1.2

Add $93$ and $112$ .

$- 2 y + 205 = 0$

$x = 31$

$z = 7$

$- 2 y + 205 = 0$

$x = 31$

$z = 7$

$- 2 y + 205 = 0$

$x = 31$

$z = 7$

$- 2 y + 205 = 0$

$x = 31$

$z = 7$

Step 3.4

Solve for $y$ in $- 2 y + 205 = 0$ .

Tap for more steps...

Step 3.4.1

Subtract $205$ from both sides of the equation.

$- 2 y = - 205$

$x = 31$

$z = 7$

Step 3.4.2

Divide each term in $- 2 y = - 205$ by $- 2$ and simplify.

Tap for more steps...

Step 3.4.2.1

Divide each term in $- 2 y = - 205$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{- 205}{- 2}$

$x = 31$

$z = 7$

Step 3.4.2.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{- 205}{- 2}$

$x = 31$

$z = 7$

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 205}{- 2}$

$x = 31$

$z = 7$

$y = \frac{- 205}{- 2}$

$x = 31$

$z = 7$

$y = \frac{- 205}{- 2}$

$x = 31$

$z = 7$

Step 3.4.2.3

Simplify the right side.

Tap for more steps...

Step 3.4.2.3.1

Dividing two negative values results in a positive value.

$y = \frac{205}{2}$

$x = 31$

$z = 7$

$y = \frac{205}{2}$

$x = 31$

$z = 7$

$y = \frac{205}{2}$

$x = 31$

$z = 7$

$y = \frac{205}{2}$

$x = 31$

$z = 7$

Step 3.5

Solve the system of equations.

$y = \frac{205}{2} x = 31 z = 7$

Step 3.6

List all of the solutions.

$y = \frac{205}{2}, x = 31, z = 7$