Solve the Matrix Equation 3x+[[-1,-2],[10,-7]]=[[-13,-2],[-5,14]]

$3 x + [\begin{array}{c} - 1 & - 2 \\ 10 & - 7 \end{array}] = [\begin{array}{c} - 13 & - 2 \\ - 5 & 14 \end{array}]$

Subtract the matrix from both sides of the equation.

$3 X + [\begin{array}{c} - 1 & - 2 \\ 10 & - 7 \end{array}] - [\begin{array}{c} - 1 & - 2 \\ 10 & - 7 \end{array}] = [\begin{array}{c} - 13 & - 2 \\ - 5 & 14 \end{array}] - [\begin{array}{c} - 1 & - 2 \\ 10 & - 7 \end{array}]$

Simplify both sides of the equation.

Simplify the left side of the equation.

Subtract the corresponding elements.

$[\begin{array}{c} - 1 - (- 1) & - 2 - (- 2) \\ 10 - 10 & - 7 - (- 7) \end{array}] + 3 X = [\begin{array}{c} - 13 & - 2 \\ - 5 & 14 \end{array}] - [\begin{array}{c} - 1 & - 2 \\ 10 & - 7 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} - 1 - (- 1) & - 2 - (- 2) \\ 10 - 10 & - 7 - (- 7) \end{array}]$ .

Simplify $- 1 - (- 1)$ .

$[\begin{array}{c} 0 & - 2 - (- 2) \\ 10 - 10 & - 7 - (- 7) \end{array}] + 3 X = [\begin{array}{c} - 13 & - 2 \\ - 5 & 14 \end{array}] - [\begin{array}{c} - 1 & - 2 \\ 10 & - 7 \end{array}]$

Simplify $- 2 - (- 2)$ .

$[\begin{array}{c} 0 & 0 \\ 10 - 10 & - 7 - (- 7) \end{array}] + 3 X = [\begin{array}{c} - 13 & - 2 \\ - 5 & 14 \end{array}] - [\begin{array}{c} - 1 & - 2 \\ 10 & - 7 \end{array}]$

Simplify $10 - 10$ .

$[\begin{array}{c} 0 & 0 \\ 0 & - 7 - (- 7) \end{array}] + 3 X = [\begin{array}{c} - 13 & - 2 \\ - 5 & 14 \end{array}] - [\begin{array}{c} - 1 & - 2 \\ 10 & - 7 \end{array}]$

Simplify $- 7 - (- 7)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 3 X = [\begin{array}{c} - 13 & - 2 \\ - 5 & 14 \end{array}] - [\begin{array}{c} - 1 & - 2 \\ 10 & - 7 \end{array}]$

Simplify the right side of the equation.

Subtract the corresponding elements.

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 3 X = [\begin{array}{c} - 13 - (- 1) & - 2 - (- 2) \\ - 5 - 10 & 14 - (- 7) \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} - 13 - (- 1) & - 2 - (- 2) \\ - 5 - 10 & 14 - (- 7) \end{array}]$ .

Simplify $- 13 - (- 1)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 3 X = [\begin{array}{c} - 12 & - 2 - (- 2) \\ - 5 - 10 & 14 - (- 7) \end{array}]$

Simplify $- 2 - (- 2)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 3 X = [\begin{array}{c} - 12 & 0 \\ - 5 - 10 & 14 - (- 7) \end{array}]$

Simplify $- 5 - 10$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 3 X = [\begin{array}{c} - 12 & 0 \\ - 15 & 14 - (- 7) \end{array}]$

Simplify $14 - (- 7)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 3 X = [\begin{array}{c} - 12 & 0 \\ - 15 & 21 \end{array}]$

Adding the null matrix to any matrix is a matrix itself.

$3 X = [\begin{array}{c} - 12 & 0 \\ - 15 & 21 \end{array}]$

Multiply both sides of the equation by $\frac{1}{3}$ .

$\frac{1}{3} \cdot (3 X) = \frac{1}{3} \cdot [\begin{array}{c} - 12 & 0 \\ - 15 & 21 \end{array}]$

Simplify both sides of the equation.

Cancel the common factor of $3$ .

Factor $3$ out of $3 X$ .

$\frac{1}{3} \cdot (3 (X)) = \frac{1}{3} \cdot [\begin{array}{c} - 12 & 0 \\ - 15 & 21 \end{array}]$

Cancel the common factor.

$\frac{1}{3} \cdot (3 X) = \frac{1}{3} \cdot [\begin{array}{c} - 12 & 0 \\ - 15 & 21 \end{array}]$

Rewrite the expression.

$X = \frac{1}{3} \cdot [\begin{array}{c} - 12 & 0 \\ - 15 & 21 \end{array}]$

Simplify the right side of the equation.

Multiply $\frac{1}{3}$ by each element of the matrix.

$X = [\begin{array}{c} \frac{1}{3} \cdot - 12 & \frac{1}{3} \cdot 0 \\ \frac{1}{3} \cdot - 15 & \frac{1}{3} \cdot 21 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{3} \cdot - 12$ .

$X = [\begin{array}{c} - 4 & \frac{1}{3} \cdot 0 \\ \frac{1}{3} \cdot - 15 & \frac{1}{3} \cdot 21 \end{array}]$

Rearrange $\frac{1}{3} \cdot 0$ .

$X = [\begin{array}{c} - 4 & 0 \\ \frac{1}{3} \cdot - 15 & \frac{1}{3} \cdot 21 \end{array}]$

Rearrange $\frac{1}{3} \cdot - 15$ .

$X = [\begin{array}{c} - 4 & 0 \\ - 5 & \frac{1}{3} \cdot 21 \end{array}]$

Rearrange $\frac{1}{3} \cdot 21$ .

$X = [\begin{array}{c} - 4 & 0 \\ - 5 & 7 \end{array}]$

Solve the Matrix Equation 3x+[[-1,-2],[10,-7]]=[[-13,-2],[-5,14]]

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