Solve the Matrix Equation [[2,5],[3,7]][[x],[y]]=[[2],[4]]

$[\begin{array}{c} 2 & 5 \\ 3 & 7 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 2 \\ 4 \end{array}]$

Step 1

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

$[\begin{array}{c} 2 x + 5 y \\ 3 x + 7 y \end{array}] = [\begin{array}{c} 2 \\ 4 \end{array}]$

Step 2

Write as a linear system of equations.

$2 x + 5 y = 2 3 x + 7 y = 4$

Step 3

Solve the system of equations.

Solve for $x$ in $2 x + 5 y = 2$ .

Subtract $5 y$ from both sides of the equation.

$2 x = 2 - 5 y$

$3 x + 7 y = 4$

Divide each term in $2 x = 2 - 5 y$ by $2$ and simplify.

Divide each term in $2 x = 2 - 5 y$ by $2$ .

$\frac{2 x}{2} = \frac{2}{2} + \frac{- 5 y}{2}$

$3 x + 7 y = 4$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{2}{2} + \frac{- 5 y}{2}$

$3 x + 7 y = 4$

Divide $x$ by $1$ .

$x = \frac{2}{2} + \frac{- 5 y}{2}$

$3 x + 7 y = 4$

$x = \frac{2}{2} + \frac{- 5 y}{2}$

$3 x + 7 y = 4$

$x = \frac{2}{2} + \frac{- 5 y}{2}$

$3 x + 7 y = 4$

Simplify the right side.

Simplify each term.

Divide $2$ by $2$ .

$x = 1 + \frac{- 5 y}{2}$

$3 x + 7 y = 4$

Move the negative in front of the fraction.

$x = 1 - \frac{5 y}{2}$

$3 x + 7 y = 4$

$x = 1 - \frac{5 y}{2}$

$3 x + 7 y = 4$

$x = 1 - \frac{5 y}{2}$

$3 x + 7 y = 4$

$x = 1 - \frac{5 y}{2}$

$3 x + 7 y = 4$

$x = 1 - \frac{5 y}{2}$

$3 x + 7 y = 4$

Replace all occurrences of $x$ with $1 - \frac{5 y}{2}$ in each equation.

Replace all occurrences of $x$ in $3 x + 7 y = 4$ with $1 - \frac{5 y}{2}$ .

$3 (1 - \frac{5 y}{2}) + 7 y = 4$

$x = 1 - \frac{5 y}{2}$

Simplify the left side.

Simplify $3 (1 - \frac{5 y}{2}) + 7 y$ .

Simplify each term.

Apply the distributive property.

$3 \cdot 1 + 3 (- \frac{5 y}{2}) + 7 y = 4$

$x = 1 - \frac{5 y}{2}$

Multiply $3$ by $1$ .

$3 + 3 (- \frac{5 y}{2}) + 7 y = 4$

$x = 1 - \frac{5 y}{2}$

Multiply $3 (- \frac{5 y}{2})$ .

Multiply $- 1$ by $3$ .

$3 - 3 \frac{5 y}{2} + 7 y = 4$

$x = 1 - \frac{5 y}{2}$

Combine $- 3$ and $\frac{5 y}{2}$ .

$3 + \frac{- 3 (5 y)}{2} + 7 y = 4$

$x = 1 - \frac{5 y}{2}$

Multiply $5$ by $- 3$ .

$3 + \frac{- 15 y}{2} + 7 y = 4$

$x = 1 - \frac{5 y}{2}$

$3 + \frac{- 15 y}{2} + 7 y = 4$

$x = 1 - \frac{5 y}{2}$

Move the negative in front of the fraction.

$3 - \frac{15 y}{2} + 7 y = 4$

$x = 1 - \frac{5 y}{2}$

$3 - \frac{15 y}{2} + 7 y = 4$

$x = 1 - \frac{5 y}{2}$

To write $7 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 - \frac{15 y}{2} + 7 y \cdot \frac{2}{2} = 4$

$x = 1 - \frac{5 y}{2}$

Simplify terms.

Combine $7 y$ and $\frac{2}{2}$ .

$3 - \frac{15 y}{2} + \frac{7 y \cdot 2}{2} = 4$

$x = 1 - \frac{5 y}{2}$

Combine the numerators over the common denominator.

$3 + \frac{- 15 y + 7 y \cdot 2}{2} = 4$

$x = 1 - \frac{5 y}{2}$

$3 + \frac{- 15 y + 7 y \cdot 2}{2} = 4$

$x = 1 - \frac{5 y}{2}$

Simplify each term.

Simplify the numerator.

Factor $y$ out of $- 15 y + 7 y \cdot 2$ .

Factor $y$ out of $- 15 y$ .

$3 + \frac{y \cdot - 15 + 7 y \cdot 2}{2} = 4$

$x = 1 - \frac{5 y}{2}$

Factor $y$ out of $7 y \cdot 2$ .

$3 + \frac{y \cdot - 15 + y (7 \cdot 2)}{2} = 4$

$x = 1 - \frac{5 y}{2}$

Factor $y$ out of $y \cdot - 15 + y (7 \cdot 2)$ .

$3 + \frac{y (- 15 + 7 \cdot 2)}{2} = 4$

$x = 1 - \frac{5 y}{2}$

$3 + \frac{y (- 15 + 7 \cdot 2)}{2} = 4$

$x = 1 - \frac{5 y}{2}$

Multiply $7$ by $2$ .

$3 + \frac{y (- 15 + 14)}{2} = 4$

$x = 1 - \frac{5 y}{2}$

Add $- 15$ and $14$ .

$3 + \frac{y \cdot - 1}{2} = 4$

$x = 1 - \frac{5 y}{2}$

$3 + \frac{y \cdot - 1}{2} = 4$

$x = 1 - \frac{5 y}{2}$

Move $- 1$ to the left of $y$ .

$3 + \frac{- 1 \cdot y}{2} = 4$

$x = 1 - \frac{5 y}{2}$

Move the negative in front of the fraction.

$3 - \frac{y}{2} = 4$

$x = 1 - \frac{5 y}{2}$

$3 - \frac{y}{2} = 4$

$x = 1 - \frac{5 y}{2}$

$3 - \frac{y}{2} = 4$

$x = 1 - \frac{5 y}{2}$

$3 - \frac{y}{2} = 4$

$x = 1 - \frac{5 y}{2}$

$3 - \frac{y}{2} = 4$

$x = 1 - \frac{5 y}{2}$

Solve for $y$ in $3 - \frac{y}{2} = 4$ .

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

Subtract $3$ from both sides of the equation.

$- \frac{y}{2} = 4 - 3$

$x = 1 - \frac{5 y}{2}$

Subtract $3$ from $4$ .

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

$x = 1 - \frac{5 y}{2}$

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

$x = 1 - \frac{5 y}{2}$

Multiply both sides of the equation by $- 2$ .

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

$x = 1 - \frac{5 y}{2}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 2 (- \frac{y}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{y}{2}$ into the numerator.

$- 2 \frac{- y}{2} = - 2 \cdot 1$

$x = 1 - \frac{5 y}{2}$

Factor $2$ out of $- 2$ .

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

$x = 1 - \frac{5 y}{2}$

Cancel the common factor.

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

$x = 1 - \frac{5 y}{2}$

Rewrite the expression.

$y = - 2 \cdot 1$

$x = 1 - \frac{5 y}{2}$

$y = - 2 \cdot 1$

$x = 1 - \frac{5 y}{2}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - 2 \cdot 1$

$x = 1 - \frac{5 y}{2}$

Multiply $y$ by $1$ .

$y = - 2 \cdot 1$

$x = 1 - \frac{5 y}{2}$

$y = - 2 \cdot 1$

$x = 1 - \frac{5 y}{2}$

$y = - 2 \cdot 1$

$x = 1 - \frac{5 y}{2}$

$y = - 2 \cdot 1$

$x = 1 - \frac{5 y}{2}$

Simplify the right side.

Multiply $- 2$ by $1$ .

$y = - 2$

$x = 1 - \frac{5 y}{2}$

$y = - 2$

$x = 1 - \frac{5 y}{2}$

$y = - 2$

$x = 1 - \frac{5 y}{2}$

$y = - 2$

$x = 1 - \frac{5 y}{2}$

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

Replace all occurrences of $y$ in $x = 1 - \frac{5 y}{2}$ with $- 2$ .

$x = 1 - \frac{5 (- 2)}{2}$

$y = - 2$

Simplify the right side.

Simplify $1 - \frac{5 (- 2)}{2}$ .

Simplify each term.

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

Factor $2$ out of $5 (- 2)$ .

$x = 1 - \frac{2 (5 \cdot (- 1))}{2}$

$y = - 2$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = 1 - \frac{2 (5 \cdot (- 1))}{2 (1)}$

$y = - 2$

Cancel the common factor.

$x = 1 - \frac{2 (5 \cdot (- 1))}{2 \cdot 1}$

$y = - 2$

Rewrite the expression.

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

$y = - 2$

Divide $5 \cdot (- 1)$ by $1$ .

$x = 1 - (5 \cdot (- 1))$

$y = - 2$

$x = 1 - (5 \cdot (- 1))$

$y = - 2$

$x = 1 - (5 \cdot (- 1))$

$y = - 2$

Multiply $- (5 \cdot (- 1))$ .

Multiply $5$ by $- 1$ .

$x = 1 + 5$

$y = - 2$

Multiply $- 1$ by $- 5$ .

$x = 1 + 5$

$y = - 2$

$x = 1 + 5$

$y = - 2$

$x = 1 + 5$

$y = - 2$

Add $1$ and $5$ .

$x = 6$

$y = - 2$

$x = 6$

$y = - 2$

$x = 6$

$y = - 2$

$x = 6$

$y = - 2$

List all of the solutions.

$x = 6, y = - 2$

Solve the Matrix Equation [[2,5],[3,7]][[x],[y]]=[[2],[4]]

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