Solve the Matrix Equation -3y+2[[6,9],[-12,15]]=[[27,-18],[30,6]]

$- 3 y + 2 [\begin{array}{c} 6 & 9 \\ - 12 & 15 \end{array}] = [\begin{array}{c} 27 & - 18 \\ 30 & 6 \end{array}]$

Step 1

Simplify each term.

Multiply $2$ by each element of the matrix.

$- 3 y + [\begin{array}{c} 2 \cdot 6 & 2 \cdot 9 \\ 2 \cdot - 12 & 2 \cdot 15 \end{array}] = [\begin{array}{c} 27 & - 18 \\ 30 & 6 \end{array}]$

Simplify each element in the matrix.

Multiply $2$ by $6$ .

$- 3 y + [\begin{array}{c} 12 & 2 \cdot 9 \\ 2 \cdot - 12 & 2 \cdot 15 \end{array}] = [\begin{array}{c} 27 & - 18 \\ 30 & 6 \end{array}]$

Multiply $2$ by $9$ .

$- 3 y + [\begin{array}{c} 12 & 18 \\ 2 \cdot - 12 & 2 \cdot 15 \end{array}] = [\begin{array}{c} 27 & - 18 \\ 30 & 6 \end{array}]$

Multiply $2$ by $- 12$ .

$- 3 y + [\begin{array}{c} 12 & 18 \\ - 24 & 2 \cdot 15 \end{array}] = [\begin{array}{c} 27 & - 18 \\ 30 & 6 \end{array}]$

Multiply $2$ by $15$ .

$- 3 y + [\begin{array}{c} 12 & 18 \\ - 24 & 30 \end{array}] = [\begin{array}{c} 27 & - 18 \\ 30 & 6 \end{array}]$

Step 2

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

Subtract $[\begin{array}{c} 12 & 18 \\ - 24 & 30 \end{array}]$ from both sides of the equation.

$- 3 y = [\begin{array}{c} 27 & - 18 \\ 30 & 6 \end{array}] - [\begin{array}{c} 12 & 18 \\ - 24 & 30 \end{array}]$

Step 3

Simplify the right side.

Subtract the corresponding elements.

$- 3 y = [\begin{array}{c} 27 - 12 & - 18 - 18 \\ 30 + 24 & 6 - 30 \end{array}]$

Simplify each element.

Subtract $12$ from $27$ .

$- 3 y = [\begin{array}{c} 15 & - 18 - 18 \\ 30 + 24 & 6 - 30 \end{array}]$

Subtract $18$ from $- 18$ .

$- 3 y = [\begin{array}{c} 15 & - 36 \\ 30 + 24 & 6 - 30 \end{array}]$

Add $30$ and $24$ .

$- 3 y = [\begin{array}{c} 15 & - 36 \\ 54 & 6 - 30 \end{array}]$

Subtract $30$ from $6$ .

$- 3 y = [\begin{array}{c} 15 & - 36 \\ 54 & - 24 \end{array}]$

Step 4

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

$- \frac{1}{3} (- 3 y) = - \frac{1}{3} [\begin{array}{c} 15 & - 36 \\ 54 & - 24 \end{array}]$

Step 5

Simplify.

Cancel the common factor of $3$ .

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

$\frac{- 1}{3} (- 3 y) = - \frac{1}{3} [\begin{array}{c} 15 & - 36 \\ 54 & - 24 \end{array}]$

Factor $3$ out of $- 3 y$ .

$\frac{- 1}{3} (3 (- y)) = - \frac{1}{3} [\begin{array}{c} 15 & - 36 \\ 54 & - 24 \end{array}]$

Cancel the common factor.

$\frac{- 1}{3} (3 (- y)) = - \frac{1}{3} [\begin{array}{c} 15 & - 36 \\ 54 & - 24 \end{array}]$

Rewrite the expression.

$- - y = - \frac{1}{3} [\begin{array}{c} 15 & - 36 \\ 54 & - 24 \end{array}]$

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{1}{3} [\begin{array}{c} 15 & - 36 \\ 54 & - 24 \end{array}]$

Multiply $y$ by $1$ .

$y = - \frac{1}{3} [\begin{array}{c} 15 & - 36 \\ 54 & - 24 \end{array}]$

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

$y = [\begin{array}{c} - \frac{1}{3} \cdot 15 & - \frac{1}{3} \cdot - 36 \\ - \frac{1}{3} \cdot 54 & - \frac{1}{3} \cdot - 24 \end{array}]$

Simplify each element in the matrix.

Cancel the common factor of $3$ .

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

$y = [\begin{array}{c} \frac{- 1}{3} \cdot 15 & - \frac{1}{3} \cdot - 36 \\ - \frac{1}{3} \cdot 54 & - \frac{1}{3} \cdot - 24 \end{array}]$

Factor $3$ out of $15$ .

$y = [\begin{array}{c} \frac{- 1}{3} \cdot (3 (5)) & - \frac{1}{3} \cdot - 36 \\ - \frac{1}{3} \cdot 54 & - \frac{1}{3} \cdot - 24 \end{array}]$

Cancel the common factor.

$y = [\begin{array}{c} \frac{- 1}{3} \cdot (3 \cdot 5) & - \frac{1}{3} \cdot - 36 \\ - \frac{1}{3} \cdot 54 & - \frac{1}{3} \cdot - 24 \end{array}]$

Rewrite the expression.

$y = [\begin{array}{c} - 1 \cdot 5 & - \frac{1}{3} \cdot - 36 \\ - \frac{1}{3} \cdot 54 & - \frac{1}{3} \cdot - 24 \end{array}]$

Multiply $- 1$ by $5$ .

$y = [\begin{array}{c} - 5 & - \frac{1}{3} \cdot - 36 \\ - \frac{1}{3} \cdot 54 & - \frac{1}{3} \cdot - 24 \end{array}]$

Cancel the common factor of $3$ .

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

$y = [\begin{array}{c} - 5 & \frac{- 1}{3} \cdot - 36 \\ - \frac{1}{3} \cdot 54 & - \frac{1}{3} \cdot - 24 \end{array}]$

Factor $3$ out of $- 36$ .

$y = [\begin{array}{c} - 5 & \frac{- 1}{3} \cdot (3 (- 12)) \\ - \frac{1}{3} \cdot 54 & - \frac{1}{3} \cdot - 24 \end{array}]$

Cancel the common factor.

$y = [\begin{array}{c} - 5 & \frac{- 1}{3} \cdot (3 \cdot - 12) \\ - \frac{1}{3} \cdot 54 & - \frac{1}{3} \cdot - 24 \end{array}]$

Rewrite the expression.

$y = [\begin{array}{c} - 5 & - 1 \cdot - 12 \\ - \frac{1}{3} \cdot 54 & - \frac{1}{3} \cdot - 24 \end{array}]$

Multiply $- 1$ by $- 12$ .

$y = [\begin{array}{c} - 5 & 12 \\ - \frac{1}{3} \cdot 54 & - \frac{1}{3} \cdot - 24 \end{array}]$

Cancel the common factor of $3$ .

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

$y = [\begin{array}{c} - 5 & 12 \\ \frac{- 1}{3} \cdot 54 & - \frac{1}{3} \cdot - 24 \end{array}]$

Factor $3$ out of $54$ .

$y = [\begin{array}{c} - 5 & 12 \\ \frac{- 1}{3} \cdot (3 (18)) & - \frac{1}{3} \cdot - 24 \end{array}]$

Cancel the common factor.

$y = [\begin{array}{c} - 5 & 12 \\ \frac{- 1}{3} \cdot (3 \cdot 18) & - \frac{1}{3} \cdot - 24 \end{array}]$

Rewrite the expression.

$y = [\begin{array}{c} - 5 & 12 \\ - 1 \cdot 18 & - \frac{1}{3} \cdot - 24 \end{array}]$

Multiply $- 1$ by $18$ .

$y = [\begin{array}{c} - 5 & 12 \\ - 18 & - \frac{1}{3} \cdot - 24 \end{array}]$

Cancel the common factor of $3$ .

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

$y = [\begin{array}{c} - 5 & 12 \\ - 18 & \frac{- 1}{3} \cdot - 24 \end{array}]$

Factor $3$ out of $- 24$ .

$y = [\begin{array}{c} - 5 & 12 \\ - 18 & \frac{- 1}{3} \cdot (3 (- 8)) \end{array}]$

Cancel the common factor.

$y = [\begin{array}{c} - 5 & 12 \\ - 18 & \frac{- 1}{3} \cdot (3 \cdot - 8) \end{array}]$

Rewrite the expression.

$y = [\begin{array}{c} - 5 & 12 \\ - 18 & - 1 \cdot - 8 \end{array}]$

Multiply $- 1$ by $- 8$ .

$y = [\begin{array}{c} - 5 & 12 \\ - 18 & 8 \end{array}]$

Solve the Matrix Equation -3y+2[[6,9],[-12,15]]=[[27,-18],[30,6]]

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