Solve the Matrix Equation x+2[[12,-6],[-19,-7]]=[[7,-7],[-14,-40]]

$x + 2 [\begin{array}{c} 12 & - 6 \\ - 19 & - 7 \end{array}] = [\begin{array}{c} 7 & - 7 \\ - 14 & - 40 \end{array}]$

Step 1

Simplify each term.

Multiply $2$ by each element of the matrix.

$x + [\begin{array}{c} 2 \cdot 12 & 2 \cdot - 6 \\ 2 \cdot - 19 & 2 \cdot - 7 \end{array}] = [\begin{array}{c} 7 & - 7 \\ - 14 & - 40 \end{array}]$

Simplify each element in the matrix.

Multiply $2$ by $12$ .

$x + [\begin{array}{c} 24 & 2 \cdot - 6 \\ 2 \cdot - 19 & 2 \cdot - 7 \end{array}] = [\begin{array}{c} 7 & - 7 \\ - 14 & - 40 \end{array}]$

Multiply $2$ by $- 6$ .

$x + [\begin{array}{c} 24 & - 12 \\ 2 \cdot - 19 & 2 \cdot - 7 \end{array}] = [\begin{array}{c} 7 & - 7 \\ - 14 & - 40 \end{array}]$

Multiply $2$ by $- 19$ .

$x + [\begin{array}{c} 24 & - 12 \\ - 38 & 2 \cdot - 7 \end{array}] = [\begin{array}{c} 7 & - 7 \\ - 14 & - 40 \end{array}]$

Multiply $2$ by $- 7$ .

$x + [\begin{array}{c} 24 & - 12 \\ - 38 & - 14 \end{array}] = [\begin{array}{c} 7 & - 7 \\ - 14 & - 40 \end{array}]$

Step 2

Move all terms not containing a variable to the right side.

Subtract $[\begin{array}{c} 24 & - 12 \\ - 38 & - 14 \end{array}]$ from both sides of the equation.

$x = [\begin{array}{c} 7 & - 7 \\ - 14 & - 40 \end{array}] - [\begin{array}{c} 24 & - 12 \\ - 38 & - 14 \end{array}]$

Step 3

Simplify the right side.

Subtract the corresponding elements.

$x = [\begin{array}{c} 7 - 24 & - 7 + 12 \\ - 14 + 38 & - 40 + 14 \end{array}]$

Simplify each element.

Subtract $24$ from $7$ .

$x = [\begin{array}{c} - 17 & - 7 + 12 \\ - 14 + 38 & - 40 + 14 \end{array}]$

Add $- 7$ and $12$ .

$x = [\begin{array}{c} - 17 & 5 \\ - 14 + 38 & - 40 + 14 \end{array}]$

Add $- 14$ and $38$ .

$x = [\begin{array}{c} - 17 & 5 \\ 24 & - 40 + 14 \end{array}]$

Add $- 40$ and $14$ .

$x = [\begin{array}{c} - 17 & 5 \\ 24 & - 26 \end{array}]$