Solve the Matrix Equation 2[[0,1],[3,-4]]-3x=[[9,-6],[1,-2]]

$2 [\begin{array}{c} 0 & 1 \\ 3 & - 4 \end{array}] - 3 x = [\begin{array}{c} 9 & - 6 \\ 1 & - 2 \end{array}]$

Step 1

Simplify each term.

Multiply $2$ by each element of the matrix.

$[\begin{array}{c} 2 \cdot 0 & 2 \cdot 1 \\ 2 \cdot 3 & 2 \cdot - 4 \end{array}] - 3 x = [\begin{array}{c} 9 & - 6 \\ 1 & - 2 \end{array}]$

Simplify each element in the matrix.

Multiply $2$ by $0$ .

$[\begin{array}{c} 0 & 2 \cdot 1 \\ 2 \cdot 3 & 2 \cdot - 4 \end{array}] - 3 x = [\begin{array}{c} 9 & - 6 \\ 1 & - 2 \end{array}]$

Multiply $2$ by $1$ .

$[\begin{array}{c} 0 & 2 \\ 2 \cdot 3 & 2 \cdot - 4 \end{array}] - 3 x = [\begin{array}{c} 9 & - 6 \\ 1 & - 2 \end{array}]$

Multiply $2$ by $3$ .

$[\begin{array}{c} 0 & 2 \\ 6 & 2 \cdot - 4 \end{array}] - 3 x = [\begin{array}{c} 9 & - 6 \\ 1 & - 2 \end{array}]$

Multiply $2$ by $- 4$ .

$[\begin{array}{c} 0 & 2 \\ 6 & - 8 \end{array}] - 3 x = [\begin{array}{c} 9 & - 6 \\ 1 & - 2 \end{array}]$

Step 2

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

Subtract $[\begin{array}{c} 0 & 2 \\ 6 & - 8 \end{array}]$ from both sides of the equation.

$- 3 x = [\begin{array}{c} 9 & - 6 \\ 1 & - 2 \end{array}] - [\begin{array}{c} 0 & 2 \\ 6 & - 8 \end{array}]$

Step 3

Simplify the right side.

Subtract the corresponding elements.

$- 3 x = [\begin{array}{c} 9 - 0 & - 6 - 2 \\ 1 - 6 & - 2 + 8 \end{array}]$

Simplify each element.

Subtract $0$ from $9$ .

$- 3 x = [\begin{array}{c} 9 & - 6 - 2 \\ 1 - 6 & - 2 + 8 \end{array}]$

Subtract $2$ from $- 6$ .

$- 3 x = [\begin{array}{c} 9 & - 8 \\ 1 - 6 & - 2 + 8 \end{array}]$

Subtract $6$ from $1$ .

$- 3 x = [\begin{array}{c} 9 & - 8 \\ - 5 & - 2 + 8 \end{array}]$

Add $- 2$ and $8$ .

$- 3 x = [\begin{array}{c} 9 & - 8 \\ - 5 & 6 \end{array}]$

Step 4

Multiply both sides by $- \frac{1}{3}$ .

$- \frac{1}{3} (- 3 x) = - \frac{1}{3} [\begin{array}{c} 9 & - 8 \\ - 5 & 6 \end{array}]$

Step 5

Simplify.

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\frac{- 1}{3} (- 3 x) = - \frac{1}{3} [\begin{array}{c} 9 & - 8 \\ - 5 & 6 \end{array}]$

Factor $3$ out of $- 3 x$ .

$\frac{- 1}{3} (3 (- x)) = - \frac{1}{3} [\begin{array}{c} 9 & - 8 \\ - 5 & 6 \end{array}]$

Cancel the common factor.

$\frac{- 1}{3} (3 (- x)) = - \frac{1}{3} [\begin{array}{c} 9 & - 8 \\ - 5 & 6 \end{array}]$

Rewrite the expression.

$- - x = - \frac{1}{3} [\begin{array}{c} 9 & - 8 \\ - 5 & 6 \end{array}]$

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{1}{3} [\begin{array}{c} 9 & - 8 \\ - 5 & 6 \end{array}]$

Multiply $x$ by $1$ .

$x = - \frac{1}{3} [\begin{array}{c} 9 & - 8 \\ - 5 & 6 \end{array}]$

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

$x = [\begin{array}{c} - \frac{1}{3} \cdot 9 & - \frac{1}{3} \cdot - 8 \\ - \frac{1}{3} \cdot - 5 & - \frac{1}{3} \cdot 6 \end{array}]$

Simplify each element in the matrix.

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$x = [\begin{array}{c} \frac{- 1}{3} \cdot 9 & - \frac{1}{3} \cdot - 8 \\ - \frac{1}{3} \cdot - 5 & - \frac{1}{3} \cdot 6 \end{array}]$

Factor $3$ out of $9$ .

$x = [\begin{array}{c} \frac{- 1}{3} \cdot (3 (3)) & - \frac{1}{3} \cdot - 8 \\ - \frac{1}{3} \cdot - 5 & - \frac{1}{3} \cdot 6 \end{array}]$

Cancel the common factor.

$x = [\begin{array}{c} \frac{- 1}{3} \cdot (3 \cdot 3) & - \frac{1}{3} \cdot - 8 \\ - \frac{1}{3} \cdot - 5 & - \frac{1}{3} \cdot 6 \end{array}]$

Rewrite the expression.

$x = [\begin{array}{c} - 1 \cdot 3 & - \frac{1}{3} \cdot - 8 \\ - \frac{1}{3} \cdot - 5 & - \frac{1}{3} \cdot 6 \end{array}]$

Multiply $- 1$ by $3$ .

$x = [\begin{array}{c} - 3 & - \frac{1}{3} \cdot - 8 \\ - \frac{1}{3} \cdot - 5 & - \frac{1}{3} \cdot 6 \end{array}]$

Multiply $- \frac{1}{3} \cdot - 8$ .

Multiply $- 8$ by $- 1$ .

$x = [\begin{array}{c} - 3 & 8 (\frac{1}{3}) \\ - \frac{1}{3} \cdot - 5 & - \frac{1}{3} \cdot 6 \end{array}]$

Combine $8$ and $\frac{1}{3}$ .

$x = [\begin{array}{c} - 3 & \frac{8}{3} \\ - \frac{1}{3} \cdot - 5 & - \frac{1}{3} \cdot 6 \end{array}]$

Multiply $- \frac{1}{3} \cdot - 5$ .

Multiply $- 5$ by $- 1$ .

$x = [\begin{array}{c} - 3 & \frac{8}{3} \\ 5 (\frac{1}{3}) & - \frac{1}{3} \cdot 6 \end{array}]$

Combine $5$ and $\frac{1}{3}$ .

$x = [\begin{array}{c} - 3 & \frac{8}{3} \\ \frac{5}{3} & - \frac{1}{3} \cdot 6 \end{array}]$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$x = [\begin{array}{c} - 3 & \frac{8}{3} \\ \frac{5}{3} & \frac{- 1}{3} \cdot 6 \end{array}]$

Factor $3$ out of $6$ .

$x = [\begin{array}{c} - 3 & \frac{8}{3} \\ \frac{5}{3} & \frac{- 1}{3} \cdot (3 (2)) \end{array}]$

Cancel the common factor.

$x = [\begin{array}{c} - 3 & \frac{8}{3} \\ \frac{5}{3} & \frac{- 1}{3} \cdot (3 \cdot 2) \end{array}]$

Rewrite the expression.

$x = [\begin{array}{c} - 3 & \frac{8}{3} \\ \frac{5}{3} & - 1 \cdot 2 \end{array}]$

Multiply $- 1$ by $2$ .

$x = [\begin{array}{c} - 3 & \frac{8}{3} \\ \frac{5}{3} & - 2 \end{array}]$

Solve the Matrix Equation 2[[0,1],[3,-4]]-3x=[[9,-6],[1,-2]]

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