Solve Using a Matrix with Cramer's Rule 6x-5y=17 , 6x+5y=7

$6 x - 5 y = 17$ , $6 x + 5 y = 7$

Represent the system of equations in matrix format.

$[\begin{array}{c} 6 & - 5 \\ 6 & 5 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 17 \\ 7 \end{array}]$

Find the determinant of $[\begin{array}{c} 6 & - 5 \\ 6 & 5 \end{array}]$ .

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

$determinant [\begin{array}{c} 6 & - 5 \\ 6 & 5 \end{array}] = | \begin{array}{c} 6 & - 5 \\ 6 & 5 \end{array} |$

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

$D = (6) (5) - 6 \cdot - 5$

Simplify the determinant.

Simplify each term.

Multiply $6$ by $5$ .

$D = 30 - 6 \cdot - 5$

Multiply $- 6$ by $- 5$ .

$D = 30 + 30$

Add $30$ and $30$ .

$D = 60$

Find the determinant of $[\begin{array}{c} 17 & - 5 \\ 7 & 5 \end{array}]$ .

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

$determinant [\begin{array}{c} 17 & - 5 \\ 7 & 5 \end{array}] = | \begin{array}{c} 17 & - 5 \\ 7 & 5 \end{array} |$

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

$D_{x} = (17) (5) - 7 \cdot - 5$

Simplify the determinant.

Simplify each term.

Multiply $17$ by $5$ .

$D_{x} = 85 - 7 \cdot - 5$

Multiply $- 7$ by $- 5$ .

$D_{x} = 85 + 35$

Add $85$ and $35$ .

$D_{x} = 120$

Find the determinant of $[\begin{array}{c} 6 & 17 \\ 6 & 7 \end{array}]$ .

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

$determinant [\begin{array}{c} 6 & 17 \\ 6 & 7 \end{array}] = | \begin{array}{c} 6 & 17 \\ 6 & 7 \end{array} |$

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

$D_{y} = (6) (7) - 6 \cdot 17$

Simplify the determinant.

Simplify each term.

Multiply $6$ by $7$ .

$D_{y} = 42 - 6 \cdot 17$

Multiply $- 6$ by $17$ .

$D_{y} = 42 - 102$

Subtract $102$ from $42$ .

$D_{y} = - 60$

Find the value of $x$ by Cramer\'s Rule, which states that $x = \frac{D_{x}}{D}$ . In this case, $x = \frac{120}{60}$ .

Remove parentheses.

$x = \frac{120}{60}$

Divide $120$ by $60$ .

$x = 2$

Find the value of $y$ by Cramer\'s Rule, which states that $y = \frac{D_{y}}{D}$ . In this case, $y = \frac{- 60}{60}$ .

Remove parentheses.

$y = \frac{- 60}{60}$

Divide $- 60$ by $60$ .

$y = - 1$

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

$x = 2, y = - 1$

Solve Using a Matrix with Cramer's Rule 6x-5y=17 , 6x+5y=7

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