Solve the Matrix Equation [[2,-2],[-1,3]]*[[x],[y]]=[[-18],[13]]

$[\begin{array}{c} 2 & - 2 \\ - 1 & 3 \end{array}] \cdot [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} - 18 \\ 13 \end{array}]$

Step 1

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

$[\begin{array}{c} 2 x - 2 y \\ - x + 3 y \end{array}] = [\begin{array}{c} - 18 \\ 13 \end{array}]$

Step 2

Write as a linear system of equations.

$2 x - 2 y = - 18 - x + 3 y = 13$

Step 3

Solve the system of equations.

Solve for $x$ in $2 x - 2 y = - 18$ .

Add $2 y$ to both sides of the equation.

$2 x = - 18 + 2 y$

$- x + 3 y = 13$

Divide each term in $2 x = - 18 + 2 y$ by $2$ and simplify.

Divide each term in $2 x = - 18 + 2 y$ by $2$ .

$\frac{2 x}{2} = \frac{- 18}{2} + \frac{2 y}{2}$

$- x + 3 y = 13$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 18}{2} + \frac{2 y}{2}$

$- x + 3 y = 13$

Divide $x$ by $1$ .

$x = \frac{- 18}{2} + \frac{2 y}{2}$

$- x + 3 y = 13$

$x = \frac{- 18}{2} + \frac{2 y}{2}$

$- x + 3 y = 13$

$x = \frac{- 18}{2} + \frac{2 y}{2}$

$- x + 3 y = 13$

Simplify the right side.

Simplify each term.

Divide $- 18$ by $2$ .

$x = - 9 + \frac{2 y}{2}$

$- x + 3 y = 13$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = - 9 + \frac{2 y}{2}$

$- x + 3 y = 13$

Divide $y$ by $1$ .

$x = - 9 + y$

$- x + 3 y = 13$

$x = - 9 + y$

$- x + 3 y = 13$

$x = - 9 + y$

$- x + 3 y = 13$

$x = - 9 + y$

$- x + 3 y = 13$

$x = - 9 + y$

$- x + 3 y = 13$

$x = - 9 + y$

$- x + 3 y = 13$

Replace all occurrences of $x$ with $- 9 + y$ in each equation.

Replace all occurrences of $x$ in $- x + 3 y = 13$ with $- 9 + y$ .

$- (- 9 + y) + 3 y = 13$

$x = - 9 + y$

Simplify the left side.

Simplify $- (- 9 + y) + 3 y$ .

Simplify each term.

Apply the distributive property.

$9 - y + 3 y = 13$

$x = - 9 + y$

Multiply $- 1$ by $- 9$ .

$9 - y + 3 y = 13$

$x = - 9 + y$

$9 - y + 3 y = 13$

$x = - 9 + y$

Add $- y$ and $3 y$ .

$9 + 2 y = 13$

$x = - 9 + y$

$9 + 2 y = 13$

$x = - 9 + y$

$9 + 2 y = 13$

$x = - 9 + y$

$9 + 2 y = 13$

$x = - 9 + y$

Solve for $y$ in $9 + 2 y = 13$ .

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

Subtract $9$ from both sides of the equation.

$2 y = 13 - 9$

$x = - 9 + y$

Subtract $9$ from $13$ .

$2 y = 4$

$x = - 9 + y$

$2 y = 4$

$x = - 9 + y$

Divide each term in $2 y = 4$ by $2$ and simplify.

Divide each term in $2 y = 4$ by $2$ .

$\frac{2 y}{2} = \frac{4}{2}$

$x = - 9 + y$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{4}{2}$

$x = - 9 + y$

Divide $y$ by $1$ .

$y = \frac{4}{2}$

$x = - 9 + y$

$y = \frac{4}{2}$

$x = - 9 + y$

$y = \frac{4}{2}$

$x = - 9 + y$

Simplify the right side.

Divide $4$ by $2$ .

$y = 2$

$x = - 9 + y$

$y = 2$

$x = - 9 + y$

$y = 2$

$x = - 9 + y$

$y = 2$

$x = - 9 + y$

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

Replace all occurrences of $y$ in $x = - 9 + y$ with $2$ .

$x = - 9 + 2$

$y = 2$

Simplify $x = - 9 + 2$ .

Simplify the left side.

Remove parentheses.

$x = - 9 + 2$

$y = 2$

$x = - 9 + 2$

$y = 2$

Simplify the right side.

Add $- 9$ and $2$ .

$x = - 7$

$y = 2$

$x = - 7$

$y = 2$

$x = - 7$

$y = 2$

$x = - 7$

$y = 2$

List all of the solutions.

$x = - 7, y = 2$

Solve the Matrix Equation [[2,-2],[-1,3]]*[[x],[y]]=[[-18],[13]]

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