Solve the Matrix Equation 1/7x-[[1,2],[7,-1]]=[[3,-2],[-5,4]]

$\frac{1}{7} x - [\begin{array}{c} 1 & 2 \\ 7 & - 1 \end{array}] = [\begin{array}{c} 3 & - 2 \\ - 5 & 4 \end{array}]$

Step 1

Combine $\frac{1}{7}$ and $x$ .

$\frac{x}{7} - [\begin{array}{c} 1 & 2 \\ 7 & - 1 \end{array}] = [\begin{array}{c} 3 & - 2 \\ - 5 & 4 \end{array}]$

Step 2

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

Add $[\begin{array}{c} 1 & 2 \\ 7 & - 1 \end{array}]$ to both sides of the equation.

$\frac{x}{7} = [\begin{array}{c} 3 & - 2 \\ - 5 & 4 \end{array}] + [\begin{array}{c} 1 & 2 \\ 7 & - 1 \end{array}]$

Step 3

Simplify the right side.

Add the corresponding elements.

$\frac{x}{7} = [\begin{array}{c} 3 + 1 & - 2 + 2 \\ - 5 + 7 & 4 - 1 \end{array}]$

Simplify each element.

Add $3$ and $1$ .

$\frac{x}{7} = [\begin{array}{c} 4 & - 2 + 2 \\ - 5 + 7 & 4 - 1 \end{array}]$

Add $- 2$ and $2$ .

$\frac{x}{7} = [\begin{array}{c} 4 & 0 \\ - 5 + 7 & 4 - 1 \end{array}]$

Add $- 5$ and $7$ .

$\frac{x}{7} = [\begin{array}{c} 4 & 0 \\ 2 & 4 - 1 \end{array}]$

Subtract $1$ from $4$ .

$\frac{x}{7} = [\begin{array}{c} 4 & 0 \\ 2 & 3 \end{array}]$

Step 4

Multiply both sides by $7$ .

$7 \frac{x}{7} = 7 [\begin{array}{c} 4 & 0 \\ 2 & 3 \end{array}]$

Step 5

Simplify.

Cancel the common factor of $7$ .

Cancel the common factor.

$7 \frac{x}{7} = 7 [\begin{array}{c} 4 & 0 \\ 2 & 3 \end{array}]$

Rewrite the expression.

$x = 7 [\begin{array}{c} 4 & 0 \\ 2 & 3 \end{array}]$

Multiply $7$ by each element of the matrix.

$x = [\begin{array}{c} 7 \cdot 4 & 7 \cdot 0 \\ 7 \cdot 2 & 7 \cdot 3 \end{array}]$

Simplify each element in the matrix.

Multiply $7$ by $4$ .

$x = [\begin{array}{c} 28 & 7 \cdot 0 \\ 7 \cdot 2 & 7 \cdot 3 \end{array}]$

Multiply $7$ by $0$ .

$x = [\begin{array}{c} 28 & 0 \\ 7 \cdot 2 & 7 \cdot 3 \end{array}]$

Multiply $7$ by $2$ .

$x = [\begin{array}{c} 28 & 0 \\ 14 & 7 \cdot 3 \end{array}]$

Multiply $7$ by $3$ .

$x = [\begin{array}{c} 28 & 0 \\ 14 & 21 \end{array}]$