Solve the Matrix Equation 2x-[[-2,0],[1,4]]=[[5,10],[-15,9]]

$2 x - [\begin{array}{c} - 2 & 0 \\ 1 & 4 \end{array}] = [\begin{array}{c} 5 & 10 \\ - 15 & 9 \end{array}]$

Step 1

Add the matrix to both sides of the equation.

$2 X - [\begin{array}{c} - 2 & 0 \\ 1 & 4 \end{array}] + [\begin{array}{c} - 2 & 0 \\ 1 & 4 \end{array}] = [\begin{array}{c} 5 & 10 \\ - 15 & 9 \end{array}] + [\begin{array}{c} - 2 & 0 \\ 1 & 4 \end{array}]$

Step 2

Simplify both sides of the equation.

Simplify the left side of the equation.

Add the corresponding elements.

$[\begin{array}{c} 2 - 2 & 0 + 0 \\ - 1 + 1 & - 4 + 4 \end{array}] + 2 X = [\begin{array}{c} 5 & 10 \\ - 15 & 9 \end{array}] + [\begin{array}{c} - 2 & 0 \\ 1 & 4 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 2 - 2 & 0 + 0 \\ - 1 + 1 & - 4 + 4 \end{array}]$ .

Simplify $2 - 2$ .

$[\begin{array}{c} 0 & 0 + 0 \\ - 1 + 1 & - 4 + 4 \end{array}] + 2 X = [\begin{array}{c} 5 & 10 \\ - 15 & 9 \end{array}] + [\begin{array}{c} - 2 & 0 \\ 1 & 4 \end{array}]$

Simplify $0 + 0$ .

$[\begin{array}{c} 0 & 0 \\ - 1 + 1 & - 4 + 4 \end{array}] + 2 X = [\begin{array}{c} 5 & 10 \\ - 15 & 9 \end{array}] + [\begin{array}{c} - 2 & 0 \\ 1 & 4 \end{array}]$

Simplify $- 1 + 1$ .

$[\begin{array}{c} 0 & 0 \\ 0 & - 4 + 4 \end{array}] + 2 X = [\begin{array}{c} 5 & 10 \\ - 15 & 9 \end{array}] + [\begin{array}{c} - 2 & 0 \\ 1 & 4 \end{array}]$

Simplify $- 4 + 4$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 2 X = [\begin{array}{c} 5 & 10 \\ - 15 & 9 \end{array}] + [\begin{array}{c} - 2 & 0 \\ 1 & 4 \end{array}]$

Simplify the right side of the equation.

Add the corresponding elements.

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 2 X = [\begin{array}{c} 5 - 2 & 10 + 0 \\ - 15 + 1 & 9 + 4 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 5 - 2 & 10 + 0 \\ - 15 + 1 & 9 + 4 \end{array}]$ .

Simplify $5 - 2$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 2 X = [\begin{array}{c} 3 & 10 + 0 \\ - 15 + 1 & 9 + 4 \end{array}]$

Simplify $10 + 0$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 2 X = [\begin{array}{c} 3 & 10 \\ - 15 + 1 & 9 + 4 \end{array}]$

Simplify $- 15 + 1$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 2 X = [\begin{array}{c} 3 & 10 \\ - 14 & 9 + 4 \end{array}]$

Simplify $9 + 4$ .

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

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

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

Step 3

Multiply both sides of the equation by $\frac{1}{2}$ .

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

Step 4

Simplify both sides of the equation.

Cancel the common factor of $2$ .

Factor $2$ out of $2 X$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side of the equation.

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

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

Simplify each element in the matrix.

Rearrange $\frac{1}{2} \cdot 3$ .

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

Rearrange $\frac{1}{2} \cdot 10$ .

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

Rearrange $\frac{1}{2} \cdot - 14$ .

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

Rearrange $\frac{1}{2} \cdot 13$ .

$X = [\begin{array}{c} \frac{3}{2} & 5 \\ - 7 & \frac{13}{2} \end{array}]$

Solve the Matrix Equation 2x-[[-2,0],[1,4]]=[[5,10],[-15,9]]

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