Solve the Matrix Equation [[2,1],[5,3]][[x],[y]]=[[5],[14]]

$[\begin{array}{c} 2 & 1 \\ 5 & 3 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 5 \\ 14 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 2 & 1 \\ 5 & 3 \end{array}] [\begin{array}{c} x \\ y \end{array}]$ .

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

$[\begin{array}{c} 2 x + 1 y \\ 5 x + 3 y \end{array}] = [\begin{array}{c} 5 \\ 14 \end{array}]$

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

$[\begin{array}{c} 2 x + y \\ 5 x + 3 y \end{array}] = [\begin{array}{c} 5 \\ 14 \end{array}]$

Step 2

Write as a linear system of equations.

$2 x + y = 5 5 x + 3 y = 14$

Step 3

Solve the system of equations.

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

$y = 5 - 2 x$

$5 x + 3 y = 14$

Replace all occurrences of $y$ with $5 - 2 x$ in each equation.

Replace all occurrences of $y$ in $5 x + 3 y = 14$ with $5 - 2 x$ .

$5 x + 3 (5 - 2 x) = 14$

$y = 5 - 2 x$

Simplify the left side.

Simplify $5 x + 3 (5 - 2 x)$ .

Simplify each term.

Apply the distributive property.

$5 x + 3 \cdot 5 + 3 (- 2 x) = 14$

$y = 5 - 2 x$

Multiply $3$ by $5$ .

$5 x + 15 + 3 (- 2 x) = 14$

$y = 5 - 2 x$

Multiply $- 2$ by $3$ .

$5 x + 15 - 6 x = 14$

$y = 5 - 2 x$

$5 x + 15 - 6 x = 14$

$y = 5 - 2 x$

Subtract $6 x$ from $5 x$ .

$- x + 15 = 14$

$y = 5 - 2 x$

$- x + 15 = 14$

$y = 5 - 2 x$

$- x + 15 = 14$

$y = 5 - 2 x$

$- x + 15 = 14$

$y = 5 - 2 x$

Solve for $x$ in $- x + 15 = 14$ .

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

Subtract $15$ from both sides of the equation.

$- x = 14 - 15$

$y = 5 - 2 x$

Subtract $15$ from $14$ .

$- x = - 1$

$y = 5 - 2 x$

$- x = - 1$

$y = 5 - 2 x$

Divide each term in $- x = - 1$ by $- 1$ and simplify.

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

$y = 5 - 2 x$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

$y = 5 - 2 x$

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

$y = 5 - 2 x$

$x = \frac{- 1}{- 1}$

$y = 5 - 2 x$

Simplify the right side.

Divide $- 1$ by $- 1$ .

$x = 1$

$y = 5 - 2 x$

$x = 1$

$y = 5 - 2 x$

$x = 1$

$y = 5 - 2 x$

$x = 1$

$y = 5 - 2 x$

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

Replace all occurrences of $x$ in $y = 5 - 2 x$ with $1$ .

$y = 5 - 2 \cdot 1$

$x = 1$

Simplify the right side.

Simplify $5 - 2 \cdot 1$ .

Multiply $- 2$ by $1$ .

$y = 5 - 2$

$x = 1$

Subtract $2$ from $5$ .

$y = 3$

$x = 1$

$y = 3$

$x = 1$

$y = 3$

$x = 1$

$y = 3$

$x = 1$

List all of the solutions.

$y = 3, x = 1$

Solve the Matrix Equation [[2,1],[5,3]][[x],[y]]=[[5],[14]]

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