Solve Using a Matrix with Cramer's Rule -2x=27-5y , 4y=20+2x

$- 2 x = 27 - 5 y$ , $4 y = 20 + 2 x$

Step 1

Add $5 y$ to both sides of the equation.

$- 2 x + 5 y = 27, 4 y = 20 + 2 x$

Step 2

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

$- 2 x + 5 y = 27, 4 y - 2 x = 20$

Step 3

Represent the system of equations in matrix format.

$[\begin{array}{c} - 2 & 5 \\ - 2 & 4 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 27 \\ 20 \end{array}]$

Step 4

Find the determinant of $[\begin{array}{c} - 2 & 5 \\ - 2 & 4 \end{array}]$ .

These are both valid notations for the determinant of a matrix.

$determinant [\begin{array}{c} - 2 & 5 \\ - 2 & 4 \end{array}] = | \begin{array}{c} - 2 & 5 \\ - 2 & 4 \end{array} |$

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$D = (- 2) (4) + 2 \cdot 5$

Simplify the determinant.

Simplify each term.

Multiply $- 2$ by $4$ .

$D = - 8 + 2 \cdot 5$

Multiply $2$ by $5$ .

$D = - 8 + 10$

Add $- 8$ and $10$ .

$D = 2$

Step 5

Find the determinant of $[\begin{array}{c} 27 & 5 \\ 20 & 4 \end{array}]$ .

These are both valid notations for the determinant of a matrix.

$determinant [\begin{array}{c} 27 & 5 \\ 20 & 4 \end{array}] = | \begin{array}{c} 27 & 5 \\ 20 & 4 \end{array} |$

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$D_{x} = (27) (4) - 20 \cdot 5$

Simplify the determinant.

Simplify each term.

Multiply $27$ by $4$ .

$D_{x} = 108 - 20 \cdot 5$

Multiply $- 20$ by $5$ .

$D_{x} = 108 - 100$

Subtract $100$ from $108$ .

$D_{x} = 8$

Step 6

Find the determinant of $[\begin{array}{c} - 2 & 27 \\ - 2 & 20 \end{array}]$ .

These are both valid notations for the determinant of a matrix.

$determinant [\begin{array}{c} - 2 & 27 \\ - 2 & 20 \end{array}] = | \begin{array}{c} - 2 & 27 \\ - 2 & 20 \end{array} |$

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$D_{y} = (- 2) (20) + 2 \cdot 27$

Simplify the determinant.

Simplify each term.

Multiply $- 2$ by $20$ .

$D_{y} = - 40 + 2 \cdot 27$

Multiply $2$ by $27$ .

$D_{y} = - 40 + 54$

Add $- 40$ and $54$ .

$D_{y} = 14$

Step 7

Find the value of $x$ by Cramer\'s Rule, which states that $x = \frac{D_{x}}{D}$ . In this case, $x = \frac{8}{2}$ .

Remove parentheses.

$x = \frac{8}{2}$

Divide $8$ by $2$ .

$x = 4$

Step 8

Find the value of $y$ by Cramer\'s Rule, which states that $y = \frac{D_{y}}{D}$ . In this case, $y = \frac{14}{2}$ .

Remove parentheses.

$y = \frac{14}{2}$

Divide $14$ by $2$ .

$y = 7$

Step 9

The solution to the system of equations using Cramer\'s Rule.

$x = 4, y = 7$

Step 10

Solve Using a Matrix with Cramer's Rule -2x=27-5y , 4y=20+2x

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