Find the Variables [[4,3x+1],[2y+5,2]]=[[4,-5],[y,-x]]

$[\begin{array}{c} 4 & 3 x + 1 \\ 2 y + 5 & 2 \end{array}] = [\begin{array}{c} 4 & - 5 \\ y & - x \end{array}]$

Step 1

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

$4 = 4$

$3 x + 1 = - 5$

$2 y + 5 = y$

$2 = - x$

Step 2

Solve for $x$ in $3 x + 1 = - 5$ .

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

Subtract $1$ from both sides of the equation.

$3 x = - 5 - 1$

$2 y + 5 = y$

$2 = - x$

Subtract $1$ from $- 5$ .

$3 x = - 6$

$2 y + 5 = y$

$2 = - x$

$3 x = - 6$

$2 y + 5 = y$

$2 = - x$

Divide each term in $3 x = - 6$ by $3$ and simplify.

Divide each term in $3 x = - 6$ by $3$ .

$\frac{3 x}{3} = \frac{- 6}{3}$

$2 y + 5 = y$

$2 = - x$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 6}{3}$

$2 y + 5 = y$

$2 = - x$

Divide $x$ by $1$ .

$x = \frac{- 6}{3}$

$2 y + 5 = y$

$2 = - x$

$x = \frac{- 6}{3}$

$2 y + 5 = y$

$2 = - x$

$x = \frac{- 6}{3}$

$2 y + 5 = y$

$2 = - x$

Simplify the right side.

Divide $- 6$ by $3$ .

$x = - 2$

$2 y + 5 = y$

$2 = - x$

$x = - 2$

$2 y + 5 = y$

$2 = - x$

$x = - 2$

$2 y + 5 = y$

$2 = - x$

$x = - 2$

$2 y + 5 = y$

$2 = - x$

Step 3

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

Move all terms containing $y$ to the left side of the equation.

Subtract $y$ from both sides of the equation.

$2 y + 5 - y = 0$

$x = - 2$

$2 = - x$

Subtract $y$ from $2 y$ .

$y + 5 = 0$

$x = - 2$

$2 = - x$

$y + 5 = 0$

$x = - 2$

$2 = - x$

Subtract $5$ from both sides of the equation.

$y = - 5$

$x = - 2$

$2 = - x$

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

$2 = - (- 2)$

$y = - 5$

$x = - 2$

Simplify the right side.

Multiply $- 1$ by $- 2$ .

$2 = 2$

$y = - 5$

$x = - 2$

$2 = 2$

$y = - 5$

$x = - 2$

$2 = 2$

$y = - 5$

$x = - 2$

Step 4

Remove any equations from the system that are always true.

$y = - 5$

$x = - 2$

Step 5

List all of the solutions.

$y = - 5, x = - 2$

Find the Variables [[4,3x+1],[2y+5,2]]=[[4,-5],[y,-x]]

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