Solve the Matrix Equation -3X+[[7,0,-1],[2,-3,4]]=[[10,0,8],[-19,-18,10]]

$- 3 X + [\begin{array}{c} 7 & 0 & - 1 \\ 2 & - 3 & 4 \end{array}] = [\begin{array}{c} 10 & 0 & 8 \\ - 19 & - 18 & 10 \end{array}]$

Step 1

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

Subtract $[\begin{array}{c} 7 & 0 & - 1 \\ 2 & - 3 & 4 \end{array}]$ from both sides of the equation.

$- 3 X = [\begin{array}{c} 10 & 0 & 8 \\ - 19 & - 18 & 10 \end{array}] - [\begin{array}{c} 7 & 0 & - 1 \\ 2 & - 3 & 4 \end{array}]$

Step 2

Simplify the right side.

Subtract the corresponding elements.

$- 3 X = [\begin{array}{c} 10 - 7 & 0 - 0 & 8 + 1 \\ - 19 - 2 & - 18 + 3 & 10 - 4 \end{array}]$

Simplify each element.

Subtract $7$ from $10$ .

$- 3 X = [\begin{array}{c} 3 & 0 - 0 & 8 + 1 \\ - 19 - 2 & - 18 + 3 & 10 - 4 \end{array}]$

Subtract $0$ from $0$ .

$- 3 X = [\begin{array}{c} 3 & 0 & 8 + 1 \\ - 19 - 2 & - 18 + 3 & 10 - 4 \end{array}]$

Add $8$ and $1$ .

$- 3 X = [\begin{array}{c} 3 & 0 & 9 \\ - 19 - 2 & - 18 + 3 & 10 - 4 \end{array}]$

Subtract $2$ from $- 19$ .

$- 3 X = [\begin{array}{c} 3 & 0 & 9 \\ - 21 & - 18 + 3 & 10 - 4 \end{array}]$

Add $- 18$ and $3$ .

$- 3 X = [\begin{array}{c} 3 & 0 & 9 \\ - 21 & - 15 & 10 - 4 \end{array}]$

Subtract $4$ from $10$ .

$- 3 X = [\begin{array}{c} 3 & 0 & 9 \\ - 21 & - 15 & 6 \end{array}]$

Step 3

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

$- \frac{1}{3} (- 3 X) = - \frac{1}{3} [\begin{array}{c} 3 & 0 & 9 \\ - 21 & - 15 & 6 \end{array}]$

Step 4

Simplify.

Cancel the common factor of $3$ .

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

$\frac{- 1}{3} (- 3 X) = - \frac{1}{3} [\begin{array}{c} 3 & 0 & 9 \\ - 21 & - 15 & 6 \end{array}]$

Factor $3$ out of $- 3 X$ .

$\frac{- 1}{3} (3 (- X)) = - \frac{1}{3} [\begin{array}{c} 3 & 0 & 9 \\ - 21 & - 15 & 6 \end{array}]$

Cancel the common factor.

$\frac{- 1}{3} (3 (- X)) = - \frac{1}{3} [\begin{array}{c} 3 & 0 & 9 \\ - 21 & - 15 & 6 \end{array}]$

Rewrite the expression.

$- - X = - \frac{1}{3} [\begin{array}{c} 3 & 0 & 9 \\ - 21 & - 15 & 6 \end{array}]$

Multiply $- 1$ by $- 1$ .

$1 X = - \frac{1}{3} [\begin{array}{c} 3 & 0 & 9 \\ - 21 & - 15 & 6 \end{array}]$

Multiply $X$ by $1$ .

$X = - \frac{1}{3} [\begin{array}{c} 3 & 0 & 9 \\ - 21 & - 15 & 6 \end{array}]$

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

$X = [\begin{array}{c} - \frac{1}{3} \cdot 3 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 9 \\ - \frac{1}{3} \cdot - 21 & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Simplify each element in the matrix.

Cancel the common factor of $3$ .

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

$X = [\begin{array}{c} \frac{- 1}{3} \cdot 3 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 9 \\ - \frac{1}{3} \cdot - 21 & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Cancel the common factor.

$X = [\begin{array}{c} \frac{- 1}{3} \cdot 3 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 9 \\ - \frac{1}{3} \cdot - 21 & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Rewrite the expression.

$X = [\begin{array}{c} - 1 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 9 \\ - \frac{1}{3} \cdot - 21 & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Multiply $- \frac{1}{3} \cdot 0$ .

Multiply $0$ by $- 1$ .

$X = [\begin{array}{c} - 1 & 0 (\frac{1}{3}) & - \frac{1}{3} \cdot 9 \\ - \frac{1}{3} \cdot - 21 & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Multiply $0$ by $\frac{1}{3}$ .

$X = [\begin{array}{c} - 1 & 0 & - \frac{1}{3} \cdot 9 \\ - \frac{1}{3} \cdot - 21 & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Cancel the common factor of $3$ .

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

$X = [\begin{array}{c} - 1 & 0 & \frac{- 1}{3} \cdot 9 \\ - \frac{1}{3} \cdot - 21 & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Factor $3$ out of $9$ .

$X = [\begin{array}{c} - 1 & 0 & \frac{- 1}{3} \cdot (3 (3)) \\ - \frac{1}{3} \cdot - 21 & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Cancel the common factor.

$X = [\begin{array}{c} - 1 & 0 & \frac{- 1}{3} \cdot (3 \cdot 3) \\ - \frac{1}{3} \cdot - 21 & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Rewrite the expression.

$X = [\begin{array}{c} - 1 & 0 & - 1 \cdot 3 \\ - \frac{1}{3} \cdot - 21 & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Multiply $- 1$ by $3$ .

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ - \frac{1}{3} \cdot - 21 & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Cancel the common factor of $3$ .

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

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ \frac{- 1}{3} \cdot - 21 & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Factor $3$ out of $- 21$ .

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ \frac{- 1}{3} \cdot (3 (- 7)) & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Cancel the common factor.

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ \frac{- 1}{3} \cdot (3 \cdot - 7) & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Rewrite the expression.

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ - 1 \cdot - 7 & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Multiply $- 1$ by $- 7$ .

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ 7 & - \frac{1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Cancel the common factor of $3$ .

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

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ 7 & \frac{- 1}{3} \cdot - 15 & - \frac{1}{3} \cdot 6 \end{array}]$

Factor $3$ out of $- 15$ .

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ 7 & \frac{- 1}{3} \cdot (3 (- 5)) & - \frac{1}{3} \cdot 6 \end{array}]$

Cancel the common factor.

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ 7 & \frac{- 1}{3} \cdot (3 \cdot - 5) & - \frac{1}{3} \cdot 6 \end{array}]$

Rewrite the expression.

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ 7 & - 1 \cdot - 5 & - \frac{1}{3} \cdot 6 \end{array}]$

Multiply $- 1$ by $- 5$ .

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ 7 & 5 & - \frac{1}{3} \cdot 6 \end{array}]$

Cancel the common factor of $3$ .

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

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ 7 & 5 & \frac{- 1}{3} \cdot 6 \end{array}]$

Factor $3$ out of $6$ .

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ 7 & 5 & \frac{- 1}{3} \cdot (3 (2)) \end{array}]$

Cancel the common factor.

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ 7 & 5 & \frac{- 1}{3} \cdot (3 \cdot 2) \end{array}]$

Rewrite the expression.

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ 7 & 5 & - 1 \cdot 2 \end{array}]$

Multiply $- 1$ by $2$ .

$X = [\begin{array}{c} - 1 & 0 & - 3 \\ 7 & 5 & - 2 \end{array}]$

Solve the Matrix Equation -3X+[[7,0,-1],[2,-3,4]]=[[10,0,8],[-19,-18,10]]

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