Find the Variables [[2,3],[1,2]][[x],[y]]=[[12],[7]]

$[\begin{array}{c} 2 & 3 \\ 1 & 2 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 12 \\ 7 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 2 & 3 \\ 1 & 2 \end{array}] [\begin{array}{c} x \\ y \end{array}]$ .

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 2 x + 3 y \\ 1 x + 2 y \end{array}] = [\begin{array}{c} 12 \\ 7 \end{array}]$

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} 2 x + 3 y \\ x + 2 y \end{array}] = [\begin{array}{c} 12 \\ 7 \end{array}]$

Step 2

The matrix equation can be written as a set of equations.

$2 x + 3 y = 12$

$x + 2 y = 7$

Step 3

Subtract $2 y$ from both sides of the equation.

$x = 7 - 2 y$

$2 x + 3 y = 12$

Step 4

Replace all occurrences of $x$ with $7 - 2 y$ in each equation.

Replace all occurrences of $x$ in $2 x + 3 y = 12$ with $7 - 2 y$ .

$2 (7 - 2 y) + 3 y = 12$

$x = 7 - 2 y$

Simplify the left side.

Simplify $2 (7 - 2 y) + 3 y$ .

Simplify each term.

Apply the distributive property.

$2 \cdot 7 + 2 (- 2 y) + 3 y = 12$

$x = 7 - 2 y$

Multiply $2$ by $7$ .

$14 + 2 (- 2 y) + 3 y = 12$

$x = 7 - 2 y$

Multiply $- 2$ by $2$ .

$14 - 4 y + 3 y = 12$

$x = 7 - 2 y$

$14 - 4 y + 3 y = 12$

$x = 7 - 2 y$

Add $- 4 y$ and $3 y$ .

$14 - y = 12$

$x = 7 - 2 y$

$14 - y = 12$

$x = 7 - 2 y$

$14 - y = 12$

$x = 7 - 2 y$

$14 - y = 12$

$x = 7 - 2 y$

Step 5

Solve for $y$ in $14 - y = 12$ .

Move all terms not containing $y$ to the right side of the equation.

Subtract $14$ from both sides of the equation.

$- y = 12 - 14$

$x = 7 - 2 y$

Subtract $14$ from $12$ .

$- y = - 2$

$x = 7 - 2 y$

$- y = - 2$

$x = 7 - 2 y$

Divide each term in $- y = - 2$ by $- 1$ and simplify.

Divide each term in $- y = - 2$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 2}{- 1}$

$x = 7 - 2 y$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 2}{- 1}$

$x = 7 - 2 y$

Divide $y$ by $1$ .

$y = \frac{- 2}{- 1}$

$x = 7 - 2 y$

$y = \frac{- 2}{- 1}$

$x = 7 - 2 y$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$y = 2$

$x = 7 - 2 y$

$y = 2$

$x = 7 - 2 y$

$y = 2$

$x = 7 - 2 y$

$y = 2$

$x = 7 - 2 y$

Step 6

Replace all occurrences of $y$ with $2$ in each equation.

Replace all occurrences of $y$ in $x = 7 - 2 y$ with $2$ .

$x = 7 - 2 \cdot 2$

$y = 2$

Simplify the right side.

Simplify $7 - 2 \cdot 2$ .

Multiply $- 2$ by $2$ .

$x = 7 - 4$

$y = 2$

Subtract $4$ from $7$ .

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

Step 7

List all of the solutions.

$x = 3, y = 2$

Find the Variables [[2,3],[1,2]][[x],[y]]=[[12],[7]]

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