Solve the Matrix Equation 8x-[[18,5],[-12,10],[18,60]]=[[22,-13],[-20,30],[54,4]]

$8 x - [\begin{array}{c} 18 & 5 \\ - 12 & 10 \\ 18 & 60 \end{array}] = [\begin{array}{c} 22 & - 13 \\ - 20 & 30 \\ 54 & 4 \end{array}]$

Step 1

Add the matrix to both sides of the equation.

$8 X - [\begin{array}{c} 18 & 5 \\ - 12 & 10 \\ 18 & 60 \end{array}] + [\begin{array}{c} 18 & 5 \\ - 12 & 10 \\ 18 & 60 \end{array}] = [\begin{array}{c} 22 & - 13 \\ - 20 & 30 \\ 54 & 4 \end{array}] + [\begin{array}{c} 18 & 5 \\ - 12 & 10 \\ 18 & 60 \end{array}]$

Step 2

Simplify both sides of the equation.

Simplify the left side of the equation.

Add the corresponding elements.

$[\begin{array}{c} - 18 + 18 & - 5 + 5 \\ 12 - 12 & - 10 + 10 \\ - 18 + 18 & - 60 + 60 \end{array}] + 8 X = [\begin{array}{c} 22 & - 13 \\ - 20 & 30 \\ 54 & 4 \end{array}] + [\begin{array}{c} 18 & 5 \\ - 12 & 10 \\ 18 & 60 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} - 18 + 18 & - 5 + 5 \\ 12 - 12 & - 10 + 10 \\ - 18 + 18 & - 60 + 60 \end{array}]$ .

Simplify $- 18 + 18$ .

$[\begin{array}{c} 0 & - 5 + 5 \\ 12 - 12 & - 10 + 10 \\ - 18 + 18 & - 60 + 60 \end{array}] + 8 X = [\begin{array}{c} 22 & - 13 \\ - 20 & 30 \\ 54 & 4 \end{array}] + [\begin{array}{c} 18 & 5 \\ - 12 & 10 \\ 18 & 60 \end{array}]$

Simplify $- 5 + 5$ .

$[\begin{array}{c} 0 & 0 \\ 12 - 12 & - 10 + 10 \\ - 18 + 18 & - 60 + 60 \end{array}] + 8 X = [\begin{array}{c} 22 & - 13 \\ - 20 & 30 \\ 54 & 4 \end{array}] + [\begin{array}{c} 18 & 5 \\ - 12 & 10 \\ 18 & 60 \end{array}]$

Simplify $12 - 12$ .

$[\begin{array}{c} 0 & 0 \\ 0 & - 10 + 10 \\ - 18 + 18 & - 60 + 60 \end{array}] + 8 X = [\begin{array}{c} 22 & - 13 \\ - 20 & 30 \\ 54 & 4 \end{array}] + [\begin{array}{c} 18 & 5 \\ - 12 & 10 \\ 18 & 60 \end{array}]$

Simplify $- 10 + 10$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ - 18 + 18 & - 60 + 60 \end{array}] + 8 X = [\begin{array}{c} 22 & - 13 \\ - 20 & 30 \\ 54 & 4 \end{array}] + [\begin{array}{c} 18 & 5 \\ - 12 & 10 \\ 18 & 60 \end{array}]$

Simplify $- 18 + 18$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & - 60 + 60 \end{array}] + 8 X = [\begin{array}{c} 22 & - 13 \\ - 20 & 30 \\ 54 & 4 \end{array}] + [\begin{array}{c} 18 & 5 \\ - 12 & 10 \\ 18 & 60 \end{array}]$

Simplify $- 60 + 60$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 8 X = [\begin{array}{c} 22 & - 13 \\ - 20 & 30 \\ 54 & 4 \end{array}] + [\begin{array}{c} 18 & 5 \\ - 12 & 10 \\ 18 & 60 \end{array}]$

Simplify the right side of the equation.

Add the corresponding elements.

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 8 X = [\begin{array}{c} 22 + 18 & - 13 + 5 \\ - 20 - 12 & 30 + 10 \\ 54 + 18 & 4 + 60 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 22 + 18 & - 13 + 5 \\ - 20 - 12 & 30 + 10 \\ 54 + 18 & 4 + 60 \end{array}]$ .

Simplify $22 + 18$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 8 X = [\begin{array}{c} 40 & - 13 + 5 \\ - 20 - 12 & 30 + 10 \\ 54 + 18 & 4 + 60 \end{array}]$

Simplify $- 13 + 5$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 8 X = [\begin{array}{c} 40 & - 8 \\ - 20 - 12 & 30 + 10 \\ 54 + 18 & 4 + 60 \end{array}]$

Simplify $- 20 - 12$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 8 X = [\begin{array}{c} 40 & - 8 \\ - 32 & 30 + 10 \\ 54 + 18 & 4 + 60 \end{array}]$

Simplify $30 + 10$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 8 X = [\begin{array}{c} 40 & - 8 \\ - 32 & 40 \\ 54 + 18 & 4 + 60 \end{array}]$

Simplify $54 + 18$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 8 X = [\begin{array}{c} 40 & - 8 \\ - 32 & 40 \\ 72 & 4 + 60 \end{array}]$

Simplify $4 + 60$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 8 X = [\begin{array}{c} 40 & - 8 \\ - 32 & 40 \\ 72 & 64 \end{array}]$

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

$8 X = [\begin{array}{c} 40 & - 8 \\ - 32 & 40 \\ 72 & 64 \end{array}]$

Step 3

Multiply both sides of the equation by $\frac{1}{8}$ .

$\frac{1}{8} \cdot (8 X) = \frac{1}{8} \cdot [\begin{array}{c} 40 & - 8 \\ - 32 & 40 \\ 72 & 64 \end{array}]$

Step 4

Simplify both sides of the equation.

Cancel the common factor of $8$ .

Factor $8$ out of $8 X$ .

$\frac{1}{8} \cdot (8 (X)) = \frac{1}{8} \cdot [\begin{array}{c} 40 & - 8 \\ - 32 & 40 \\ 72 & 64 \end{array}]$

Cancel the common factor.

$\frac{1}{8} \cdot (8 X) = \frac{1}{8} \cdot [\begin{array}{c} 40 & - 8 \\ - 32 & 40 \\ 72 & 64 \end{array}]$

Rewrite the expression.

$X = \frac{1}{8} \cdot [\begin{array}{c} 40 & - 8 \\ - 32 & 40 \\ 72 & 64 \end{array}]$

Simplify the right side of the equation.

Multiply $\frac{1}{8}$ by each element of the matrix.

$X = [\begin{array}{c} \frac{1}{8} \cdot 40 & \frac{1}{8} \cdot - 8 \\ \frac{1}{8} \cdot - 32 & \frac{1}{8} \cdot 40 \\ \frac{1}{8} \cdot 72 & \frac{1}{8} \cdot 64 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{8} \cdot 40$ .

$X = [\begin{array}{c} 5 & \frac{1}{8} \cdot - 8 \\ \frac{1}{8} \cdot - 32 & \frac{1}{8} \cdot 40 \\ \frac{1}{8} \cdot 72 & \frac{1}{8} \cdot 64 \end{array}]$

Rearrange $\frac{1}{8} \cdot - 8$ .

$X = [\begin{array}{c} 5 & - 1 \\ \frac{1}{8} \cdot - 32 & \frac{1}{8} \cdot 40 \\ \frac{1}{8} \cdot 72 & \frac{1}{8} \cdot 64 \end{array}]$

Rearrange $\frac{1}{8} \cdot - 32$ .

$X = [\begin{array}{c} 5 & - 1 \\ - 4 & \frac{1}{8} \cdot 40 \\ \frac{1}{8} \cdot 72 & \frac{1}{8} \cdot 64 \end{array}]$

Rearrange $\frac{1}{8} \cdot 40$ .

$X = [\begin{array}{c} 5 & - 1 \\ - 4 & 5 \\ \frac{1}{8} \cdot 72 & \frac{1}{8} \cdot 64 \end{array}]$

Rearrange $\frac{1}{8} \cdot 72$ .

$X = [\begin{array}{c} 5 & - 1 \\ - 4 & 5 \\ 9 & \frac{1}{8} \cdot 64 \end{array}]$

Rearrange $\frac{1}{8} \cdot 64$ .

$X = [\begin{array}{c} 5 & - 1 \\ - 4 & 5 \\ 9 & 8 \end{array}]$

Solve the Matrix Equation 8x-[[18,5],[-12,10],[18,60]]=[[22,-13],[-20,30],[54,4]]

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