Solve the Matrix Equation [[1,1],[2,3]][[x],[y]]=[[8],[36]]

$[\begin{array}{c} 1 & 1 \\ 2 & 3 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 8 \\ 36 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 1 & 1 \\ 2 & 3 \end{array}] [\begin{array}{c} x \\ y \end{array}]$ .

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

$[\begin{array}{c} 1 x + 1 y \\ 2 x + 3 y \end{array}] = [\begin{array}{c} 8 \\ 36 \end{array}]$

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

Multiply $x$ by $1$ .

$[\begin{array}{c} x + 1 y \\ 2 x + 3 y \end{array}] = [\begin{array}{c} 8 \\ 36 \end{array}]$

Multiply $y$ by $1$ .

$[\begin{array}{c} x + y \\ 2 x + 3 y \end{array}] = [\begin{array}{c} 8 \\ 36 \end{array}]$

Step 2

Write as a linear system of equations.

$x + y = 8 2 x + 3 y = 36$

Step 3

Solve the system of equations.

Subtract $y$ from both sides of the equation.

$x = 8 - y$

$2 x + 3 y = 36$

Replace all occurrences of $x$ with $8 - y$ in each equation.

Replace all occurrences of $x$ in $2 x + 3 y = 36$ with $8 - y$ .

$2 (8 - y) + 3 y = 36$

$x = 8 - y$

Simplify the left side.

Simplify $2 (8 - y) + 3 y$ .

Simplify each term.

Apply the distributive property.

$2 \cdot 8 + 2 (- y) + 3 y = 36$

$x = 8 - y$

Multiply $2$ by $8$ .

$16 + 2 (- y) + 3 y = 36$

$x = 8 - y$

Multiply $- 1$ by $2$ .

$16 - 2 y + 3 y = 36$

$x = 8 - y$

$16 - 2 y + 3 y = 36$

$x = 8 - y$

Add $- 2 y$ and $3 y$ .

$16 + y = 36$

$x = 8 - y$

$16 + y = 36$

$x = 8 - y$

$16 + y = 36$

$x = 8 - y$

$16 + y = 36$

$x = 8 - y$

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

Subtract $16$ from both sides of the equation.

$y = 36 - 16$

$x = 8 - y$

Subtract $16$ from $36$ .

$y = 20$

$x = 8 - y$

$y = 20$

$x = 8 - y$

Replace all occurrences of $y$ with $20$ in each equation.

Replace all occurrences of $y$ in $x = 8 - y$ with $20$ .

$x = 8 - (20)$

$y = 20$

Simplify the right side.

Simplify $8 - (20)$ .

Multiply $- 1$ by $20$ .

$x = 8 - 20$

$y = 20$

Subtract $20$ from $8$ .

$x = - 12$

$y = 20$

$x = - 12$

$y = 20$

$x = - 12$

$y = 20$

$x = - 12$

$y = 20$

List all of the solutions.

$x = - 12, y = 20$

Solve the Matrix Equation [[1,1],[2,3]][[x],[y]]=[[8],[36]]

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