Find the Variables [[2x],[x]]-[[8y],[y]]=[[12],[1]]

$[\begin{array}{c} 2 x \\ x \end{array}] - [\begin{array}{c} 8 y \\ y \end{array}] = [\begin{array}{c} 12 \\ 1 \end{array}]$

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

$2 x + 8 y = 12$

$x + y = 1$

Subtract $y$ from both sides of the equation.

$x = 1 - y$

$2 x + 8 y = 12$

Replace all occurrences of $x$ with $1 - y$ in each equation.

Replace all occurrences of $x$ in $2 x + 8 y = 12$ with $1 - y$ .

$2 (1 - y) + 8 y = 12$

$x = 1 - y$

Simplify the left side.

Simplify $2 (1 - y) + 8 y$ .

Simplify each term.

Apply the distributive property.

$2 \cdot 1 + 2 (- y) + 8 y = 12$

$x = 1 - y$

Multiply $2$ by $1$ .

$2 + 2 (- y) + 8 y = 12$

$x = 1 - y$

Multiply $- 1$ by $2$ .

$2 - 2 y + 8 y = 12$

$x = 1 - y$

$2 - 2 y + 8 y = 12$

$x = 1 - y$

Add $- 2 y$ and $8 y$ .

$2 + 6 y = 12$

$x = 1 - y$

$2 + 6 y = 12$

$x = 1 - y$

$2 + 6 y = 12$

$x = 1 - y$

$2 + 6 y = 12$

$x = 1 - y$

Solve for $y$ in $2 + 6 y = 12$ .

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

Subtract $2$ from both sides of the equation.

$6 y = 12 - 2$

$x = 1 - y$

Subtract $2$ from $12$ .

$6 y = 10$

$x = 1 - y$

$6 y = 10$

$x = 1 - y$

Divide each term in $6 y = 10$ by $6$ and simplify.

Divide each term in $6 y = 10$ by $6$ .

$\frac{6 y}{6} = \frac{10}{6}$

$x = 1 - y$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{10}{6}$

$x = 1 - y$

Divide $y$ by $1$ .

$y = \frac{10}{6}$

$x = 1 - y$

$y = \frac{10}{6}$

$x = 1 - y$

$y = \frac{10}{6}$

$x = 1 - y$

Simplify the right side.

Cancel the common factor of $10$ and $6$ .

Factor $2$ out of $10$ .

$y = \frac{2 (5)}{6}$

$x = 1 - y$

Cancel the common factors.

Factor $2$ out of $6$ .

$y = \frac{2 \cdot 5}{2 \cdot 3}$

$x = 1 - y$

Cancel the common factor.

$y = \frac{2 \cdot 5}{2 \cdot 3}$

$x = 1 - y$

Rewrite the expression.

$y = \frac{5}{3}$

$x = 1 - y$

$y = \frac{5}{3}$

$x = 1 - y$

$y = \frac{5}{3}$

$x = 1 - y$

$y = \frac{5}{3}$

$x = 1 - y$

$y = \frac{5}{3}$

$x = 1 - y$

$y = \frac{5}{3}$

$x = 1 - y$

Replace all occurrences of $y$ with $\frac{5}{3}$ in each equation.

Replace all occurrences of $y$ in $x = 1 - y$ with $\frac{5}{3}$ .

$x = 1 - (\frac{5}{3})$

$y = \frac{5}{3}$

Simplify the right side.

Simplify $1 - (\frac{5}{3})$ .

Write $1$ as a fraction with a common denominator.

$x = \frac{3}{3} - \frac{5}{3}$

$y = \frac{5}{3}$

Combine the numerators over the common denominator.

$x = \frac{3 - 5}{3}$

$y = \frac{5}{3}$

Subtract $5$ from $3$ .

$x = \frac{- 2}{3}$

$y = \frac{5}{3}$

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

$y = \frac{5}{3}$

$x = - \frac{2}{3}$

$y = \frac{5}{3}$

$x = - \frac{2}{3}$

$y = \frac{5}{3}$

$x = - \frac{2}{3}$

$y = \frac{5}{3}$

List all of the solutions.

$x = - \frac{2}{3}, y = \frac{5}{3}$

Find the Variables [[2x],[x]]-[[8y],[y]]=[[12],[1]]

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