Solve the Matrix Equation -2x+2[[2,-8],[-4,2]]=[[4,-6],[2,-8]]

$- 2 x + 2 [\begin{array}{c} 2 & - 8 \\ - 4 & 2 \end{array}] = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}]$

Step 1

Simplify each term.

Multiply $2$ by each element of the matrix.

$- 2 x + [\begin{array}{c} 2 \cdot 2 & 2 \cdot - 8 \\ 2 \cdot - 4 & 2 \cdot 2 \end{array}] = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}]$

Simplify each element in the matrix.

Multiply $2$ by $2$ .

$- 2 x + [\begin{array}{c} 4 & 2 \cdot - 8 \\ 2 \cdot - 4 & 2 \cdot 2 \end{array}] = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}]$

Multiply $2$ by $- 8$ .

$- 2 x + [\begin{array}{c} 4 & - 16 \\ 2 \cdot - 4 & 2 \cdot 2 \end{array}] = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}]$

Multiply $2$ by $- 4$ .

$- 2 x + [\begin{array}{c} 4 & - 16 \\ - 8 & 2 \cdot 2 \end{array}] = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}]$

Multiply $2$ by $2$ .

$- 2 x + [\begin{array}{c} 4 & - 16 \\ - 8 & 4 \end{array}] = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}]$

Step 2

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

Subtract $[\begin{array}{c} 4 & - 16 \\ - 8 & 4 \end{array}]$ from both sides of the equation.

$- 2 x = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}] - [\begin{array}{c} 4 & - 16 \\ - 8 & 4 \end{array}]$

Step 3

Simplify the right side.

Subtract the corresponding elements.

$- 2 x = [\begin{array}{c} 4 - 4 & - 6 + 16 \\ 2 + 8 & - 8 - 4 \end{array}]$

Simplify each element.

Subtract $4$ from $4$ .

$- 2 x = [\begin{array}{c} 0 & - 6 + 16 \\ 2 + 8 & - 8 - 4 \end{array}]$

Add $- 6$ and $16$ .

$- 2 x = [\begin{array}{c} 0 & 10 \\ 2 + 8 & - 8 - 4 \end{array}]$

Add $2$ and $8$ .

$- 2 x = [\begin{array}{c} 0 & 10 \\ 10 & - 8 - 4 \end{array}]$

Subtract $4$ from $- 8$ .

$- 2 x = [\begin{array}{c} 0 & 10 \\ 10 & - 12 \end{array}]$

Step 4

Multiply both sides by $- \frac{1}{2}$ .

$- \frac{1}{2} (- 2 x) = - \frac{1}{2} [\begin{array}{c} 0 & 10 \\ 10 & - 12 \end{array}]$

Step 5

Simplify.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{- 1}{2} (- 2 x) = - \frac{1}{2} [\begin{array}{c} 0 & 10 \\ 10 & - 12 \end{array}]$

Factor $2$ out of $- 2 x$ .

$\frac{- 1}{2} (2 (- x)) = - \frac{1}{2} [\begin{array}{c} 0 & 10 \\ 10 & - 12 \end{array}]$

Cancel the common factor.

$\frac{- 1}{2} (2 (- x)) = - \frac{1}{2} [\begin{array}{c} 0 & 10 \\ 10 & - 12 \end{array}]$

Rewrite the expression.

$- - x = - \frac{1}{2} [\begin{array}{c} 0 & 10 \\ 10 & - 12 \end{array}]$

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{1}{2} [\begin{array}{c} 0 & 10 \\ 10 & - 12 \end{array}]$

Multiply $x$ by $1$ .

$x = - \frac{1}{2} [\begin{array}{c} 0 & 10 \\ 10 & - 12 \end{array}]$

Multiply $- \frac{1}{2}$ by each element of the matrix.

$x = [\begin{array}{c} - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \\ - \frac{1}{2} \cdot 10 & - \frac{1}{2} \cdot - 12 \end{array}]$

Simplify each element in the matrix.

Multiply $- \frac{1}{2} \cdot 0$ .

Multiply $0$ by $- 1$ .

$x = [\begin{array}{c} 0 (\frac{1}{2}) & - \frac{1}{2} \cdot 10 \\ - \frac{1}{2} \cdot 10 & - \frac{1}{2} \cdot - 12 \end{array}]$

Multiply $0$ by $\frac{1}{2}$ .

$x = [\begin{array}{c} 0 & - \frac{1}{2} \cdot 10 \\ - \frac{1}{2} \cdot 10 & - \frac{1}{2} \cdot - 12 \end{array}]$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$x = [\begin{array}{c} 0 & \frac{- 1}{2} \cdot 10 \\ - \frac{1}{2} \cdot 10 & - \frac{1}{2} \cdot - 12 \end{array}]$

Factor $2$ out of $10$ .

$x = [\begin{array}{c} 0 & \frac{- 1}{2} \cdot (2 (5)) \\ - \frac{1}{2} \cdot 10 & - \frac{1}{2} \cdot - 12 \end{array}]$

Cancel the common factor.

$x = [\begin{array}{c} 0 & \frac{- 1}{2} \cdot (2 \cdot 5) \\ - \frac{1}{2} \cdot 10 & - \frac{1}{2} \cdot - 12 \end{array}]$

Rewrite the expression.

$x = [\begin{array}{c} 0 & - 1 \cdot 5 \\ - \frac{1}{2} \cdot 10 & - \frac{1}{2} \cdot - 12 \end{array}]$

Multiply $- 1$ by $5$ .

$x = [\begin{array}{c} 0 & - 5 \\ - \frac{1}{2} \cdot 10 & - \frac{1}{2} \cdot - 12 \end{array}]$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$x = [\begin{array}{c} 0 & - 5 \\ \frac{- 1}{2} \cdot 10 & - \frac{1}{2} \cdot - 12 \end{array}]$

Factor $2$ out of $10$ .

$x = [\begin{array}{c} 0 & - 5 \\ \frac{- 1}{2} \cdot (2 (5)) & - \frac{1}{2} \cdot - 12 \end{array}]$

Cancel the common factor.

$x = [\begin{array}{c} 0 & - 5 \\ \frac{- 1}{2} \cdot (2 \cdot 5) & - \frac{1}{2} \cdot - 12 \end{array}]$

Rewrite the expression.

$x = [\begin{array}{c} 0 & - 5 \\ - 1 \cdot 5 & - \frac{1}{2} \cdot - 12 \end{array}]$

Multiply $- 1$ by $5$ .

$x = [\begin{array}{c} 0 & - 5 \\ - 5 & - \frac{1}{2} \cdot - 12 \end{array}]$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$x = [\begin{array}{c} 0 & - 5 \\ - 5 & \frac{- 1}{2} \cdot - 12 \end{array}]$

Factor $2$ out of $- 12$ .

$x = [\begin{array}{c} 0 & - 5 \\ - 5 & \frac{- 1}{2} \cdot (2 (- 6)) \end{array}]$

Cancel the common factor.

$x = [\begin{array}{c} 0 & - 5 \\ - 5 & \frac{- 1}{2} \cdot (2 \cdot - 6) \end{array}]$

Rewrite the expression.

$x = [\begin{array}{c} 0 & - 5 \\ - 5 & - 1 \cdot - 6 \end{array}]$

Multiply $- 1$ by $- 6$ .

$x = [\begin{array}{c} 0 & - 5 \\ - 5 & 6 \end{array}]$

Solve the Matrix Equation -2x+2[[2,-8],[-4,2]]=[[4,-6],[2,-8]]

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