Find the Variables [[2,2],[7,3]]*[[x],[y]]=[[-4],[10]]

$[\begin{array}{c} 2 & 2 \\ 7 & 3 \end{array}] \cdot [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} - 4 \\ 10 \end{array}]$

Step 1

Simplify the left side of the equation.

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

$[\begin{array}{c} 2 \cdot x + 2 \cdot y \\ 7 \cdot x + 3 \cdot y \end{array}] = [\begin{array}{c} - 4 \\ 10 \end{array}]$

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

$[\begin{array}{c} 2 x + 2 y \\ 7 x + 3 y \end{array}] = [\begin{array}{c} - 4 \\ 10 \end{array}]$

Step 2

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

$2 x + 2 y = - 4$

$7 x + 3 y = 10$

Step 3

Solve for $x$ in $2 x + 2 y = - 4$ .

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

$2 x = - 4 - 2 y$

$7 x + 3 y = 10$

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

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

$\frac{2 x}{2} = \frac{- 4}{2} + \frac{- 2 y}{2}$

$7 x + 3 y = 10$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 4}{2} + \frac{- 2 y}{2}$

$7 x + 3 y = 10$

Divide $x$ by $1$ .

$x = \frac{- 4}{2} + \frac{- 2 y}{2}$

$7 x + 3 y = 10$

$x = \frac{- 4}{2} + \frac{- 2 y}{2}$

$7 x + 3 y = 10$

$x = \frac{- 4}{2} + \frac{- 2 y}{2}$

$7 x + 3 y = 10$

Simplify the right side.

Simplify each term.

Divide $- 4$ by $2$ .

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

$7 x + 3 y = 10$

Cancel the common factor of $- 2$ and $2$ .

Factor $2$ out of $- 2 y$ .

$x = - 2 + \frac{2 (- y)}{2}$

$7 x + 3 y = 10$

Cancel the common factors.

Factor $2$ out of $2$ .

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

$7 x + 3 y = 10$

Cancel the common factor.

$x = - 2 + \frac{2 (- y)}{2 \cdot 1}$

$7 x + 3 y = 10$

Rewrite the expression.

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

$7 x + 3 y = 10$

Divide $- y$ by $1$ .

$x = - 2 - y$

$7 x + 3 y = 10$

$x = - 2 - y$

$7 x + 3 y = 10$

$x = - 2 - y$

$7 x + 3 y = 10$

$x = - 2 - y$

$7 x + 3 y = 10$

$x = - 2 - y$

$7 x + 3 y = 10$

$x = - 2 - y$

$7 x + 3 y = 10$

$x = - 2 - y$

$7 x + 3 y = 10$

Step 4

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

Replace all occurrences of $x$ in $7 x + 3 y = 10$ with $- 2 - y$ .

$7 (- 2 - y) + 3 y = 10$

$x = - 2 - y$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$7 \cdot - 2 + 7 (- y) + 3 y = 10$

$x = - 2 - y$

Multiply $7$ by $- 2$ .

$- 14 + 7 (- y) + 3 y = 10$

$x = - 2 - y$

Multiply $- 1$ by $7$ .

$- 14 - 7 y + 3 y = 10$

$x = - 2 - y$

$- 14 - 7 y + 3 y = 10$

$x = - 2 - y$

Add $- 7 y$ and $3 y$ .

$- 14 - 4 y = 10$

$x = - 2 - y$

$- 14 - 4 y = 10$

$x = - 2 - y$

$- 14 - 4 y = 10$

$x = - 2 - y$

$- 14 - 4 y = 10$

$x = - 2 - y$

Step 5

Solve for $y$ in $- 14 - 4 y = 10$ .

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

Add $14$ to both sides of the equation.

$- 4 y = 10 + 14$

$x = - 2 - y$

Add $10$ and $14$ .

$- 4 y = 24$

$x = - 2 - y$

$- 4 y = 24$

$x = - 2 - y$

Divide each term in $- 4 y = 24$ by $- 4$ and simplify.

Divide each term in $- 4 y = 24$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{24}{- 4}$

$x = - 2 - y$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{24}{- 4}$

$x = - 2 - y$

Divide $y$ by $1$ .

$y = \frac{24}{- 4}$

$x = - 2 - y$

$y = \frac{24}{- 4}$

$x = - 2 - y$

$y = \frac{24}{- 4}$

$x = - 2 - y$

Simplify the right side.

Divide $24$ by $- 4$ .

$y = - 6$

$x = - 2 - y$

$y = - 6$

$x = - 2 - y$

$y = - 6$

$x = - 2 - y$

$y = - 6$

$x = - 2 - y$

Step 6

Replace all occurrences of $y$ with $- 6$ in each equation.

Replace all occurrences of $y$ in $x = - 2 - y$ with $- 6$ .

$x = - 2 - (- 6)$

$y = - 6$

Simplify the right side.

Simplify $- 2 - (- 6)$ .

Multiply $- 1$ by $- 6$ .

$x = - 2 + 6$

$y = - 6$

Add $- 2$ and $6$ .

$x = 4$

$y = - 6$

$x = 4$

$y = - 6$

$x = 4$

$y = - 6$

$x = 4$

$y = - 6$

Step 7

List all of the solutions.

$x = 4, y = - 6$

Find the Variables [[2,2],[7,3]]*[[x],[y]]=[[-4],[10]]

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