Find the Variables [[-1,3],[4,-2]][[x],[y]]=[[2],[22]]

$[\begin{array}{c} - 1 & 3 \\ 4 & - 2 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 2 \\ 22 \end{array}]$

Simplify the left side of the equation.

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

$[\begin{array}{c} - 1 \cdot x + 3 \cdot y \\ 4 \cdot x - 2 \cdot y \end{array}] = [\begin{array}{c} 2 \\ 22 \end{array}]$

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

$[\begin{array}{c} - x + 3 y \\ 4 x - 2 y \end{array}] = [\begin{array}{c} 2 \\ 22 \end{array}]$

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

$- x + 3 y = 2$

$4 x - 2 y = 22$

Solve for $x$ in $- x + 3 y = 2$ .

Subtract $3 y$ from both sides of the equation.

$- x = 2 - 3 y$

$4 x - 2 y = 22$

Divide each term in $- x = 2 - 3 y$ by $- 1$ and simplify.

Divide each term in $- x = 2 - 3 y$ by $- 1$ .

$\frac{- x}{- 1} = \frac{2}{- 1} + \frac{- 3 y}{- 1}$

$4 x - 2 y = 22$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{2}{- 1} + \frac{- 3 y}{- 1}$

$4 x - 2 y = 22$

Divide $x$ by $1$ .

$x = \frac{2}{- 1} + \frac{- 3 y}{- 1}$

$4 x - 2 y = 22$

$x = \frac{2}{- 1} + \frac{- 3 y}{- 1}$

$4 x - 2 y = 22$

Simplify the right side.

Simplify each term.

Divide $2$ by $- 1$ .

$x = - 2 + \frac{- 3 y}{- 1}$

$4 x - 2 y = 22$

Move the negative one from the denominator of $\frac{- 3 y}{- 1}$ .

$x = - 2 - 1 \cdot (- 3 y)$

$4 x - 2 y = 22$

Rewrite $- 1 \cdot (- 3 y)$ as $- (- 3 y)$ .

$x = - 2 - (- 3 y)$

$4 x - 2 y = 22$

Multiply $- 3$ by $- 1$ .

$x = - 2 + 3 y$

$4 x - 2 y = 22$

$x = - 2 + 3 y$

$4 x - 2 y = 22$

$x = - 2 + 3 y$

$4 x - 2 y = 22$

$x = - 2 + 3 y$

$4 x - 2 y = 22$

$x = - 2 + 3 y$

$4 x - 2 y = 22$

Replace all occurrences of $x$ with $- 2 + 3 y$ in each equation.

Replace all occurrences of $x$ in $4 x - 2 y = 22$ with $- 2 + 3 y$ .

$4 (- 2 + 3 y) - 2 y = 22$

$x = - 2 + 3 y$

Simplify the left side.

Simplify $4 (- 2 + 3 y) - 2 y$ .

Simplify each term.

Apply the distributive property.

$4 \cdot - 2 + 4 (3 y) - 2 y = 22$

$x = - 2 + 3 y$

Multiply $4$ by $- 2$ .

$- 8 + 4 (3 y) - 2 y = 22$

$x = - 2 + 3 y$

Multiply $3$ by $4$ .

$- 8 + 12 y - 2 y = 22$

$x = - 2 + 3 y$

$- 8 + 12 y - 2 y = 22$

$x = - 2 + 3 y$

Subtract $2 y$ from $12 y$ .

$- 8 + 10 y = 22$

$x = - 2 + 3 y$

$- 8 + 10 y = 22$

$x = - 2 + 3 y$

$- 8 + 10 y = 22$

$x = - 2 + 3 y$

$- 8 + 10 y = 22$

$x = - 2 + 3 y$

Solve for $y$ in $- 8 + 10 y = 22$ .

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

Add $8$ to both sides of the equation.

$10 y = 22 + 8$

$x = - 2 + 3 y$

Add $22$ and $8$ .

$10 y = 30$

$x = - 2 + 3 y$

$10 y = 30$

$x = - 2 + 3 y$

Divide each term in $10 y = 30$ by $10$ and simplify.

Divide each term in $10 y = 30$ by $10$ .

$\frac{10 y}{10} = \frac{30}{10}$

$x = - 2 + 3 y$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 y}{10} = \frac{30}{10}$

$x = - 2 + 3 y$

Divide $y$ by $1$ .

$y = \frac{30}{10}$

$x = - 2 + 3 y$

$y = \frac{30}{10}$

$x = - 2 + 3 y$

$y = \frac{30}{10}$

$x = - 2 + 3 y$

Simplify the right side.

Divide $30$ by $10$ .

$y = 3$

$x = - 2 + 3 y$

$y = 3$

$x = - 2 + 3 y$

$y = 3$

$x = - 2 + 3 y$

$y = 3$

$x = - 2 + 3 y$

Replace all occurrences of $y$ with $3$ in each equation.

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

$x = - 2 + 3 (3)$

$y = 3$

Simplify the right side.

Simplify $- 2 + 3 (3)$ .

Multiply $3$ by $3$ .

$x = - 2 + 9$

$y = 3$

Add $- 2$ and $9$ .

$x = 7$

$y = 3$

$x = 7$

$y = 3$

$x = 7$

$y = 3$

$x = 7$

$y = 3$

List all of the solutions.

$x = 7, y = 3$

Find the Variables [[-1,3],[4,-2]][[x],[y]]=[[2],[22]]

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