Solve Using a Matrix with Cramer's Rule 7x+4y=5 , -x+2y=-2

$7 x + 4 y = 5$ , $- x + 2 y = - 2$

Represent the system of equations in matrix format.

$[\begin{array}{c} 7 & 4 \\ - 1 & 2 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 5 \\ - 2 \end{array}]$

Find the determinant of $[\begin{array}{c} 7 & 4 \\ - 1 & 2 \end{array}]$ .

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

$determinant [\begin{array}{c} 7 & 4 \\ - 1 & 2 \end{array}] = | \begin{array}{c} 7 & 4 \\ - 1 & 2 \end{array} |$

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

$D = (7) (2) + 1 \cdot 4$

Simplify the determinant.

Simplify each term.

Multiply $7$ by $2$ .

$D = 14 + 1 \cdot 4$

Multiply $4$ by $1$ .

$D = 14 + 4$

Add $14$ and $4$ .

$D = 18$

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

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

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

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

$D_{x} = (5) (2) + 2 \cdot 4$

Simplify the determinant.

Simplify each term.

Multiply $5$ by $2$ .

$D_{x} = 10 + 2 \cdot 4$

Multiply $2$ by $4$ .

$D_{x} = 10 + 8$

Add $10$ and $8$ .

$D_{x} = 18$

Find the determinant of $[\begin{array}{c} 7 & 5 \\ - 1 & - 2 \end{array}]$ .

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

$determinant [\begin{array}{c} 7 & 5 \\ - 1 & - 2 \end{array}] = | \begin{array}{c} 7 & 5 \\ - 1 & - 2 \end{array} |$

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

$D_{y} = (7) (- 2) + 1 \cdot 5$

Simplify the determinant.

Simplify each term.

Multiply $7$ by $- 2$ .

$D_{y} = - 14 + 1 \cdot 5$

Multiply $5$ by $1$ .

$D_{y} = - 14 + 5$

Add $- 14$ and $5$ .

$D_{y} = - 9$

Find the value of $x$ by Cramer\'s Rule, which states that $x = \frac{D_{x}}{D}$ . In this case, $x = \frac{18}{18}$ .

Remove parentheses.

$x = \frac{18}{18}$

Divide $18$ by $18$ .

$x = 1$

Find the value of $y$ by Cramer\'s Rule, which states that $y = \frac{D_{y}}{D}$ . In this case, $y = \frac{- 9}{18}$ .

Remove parentheses.

$y = \frac{- 9}{18}$

Simplify $\frac{- 9}{18}$ .

Cancel the common factor of $- 9$ and $18$ .

Factor $9$ out of $- 9$ .

$y = \frac{9 (- 1)}{18}$

Cancel the common factors.

Factor $9$ out of $18$ .

$y = \frac{9 \cdot - 1}{9 \cdot 2}$

Cancel the common factor.

$y = \frac{9 \cdot - 1}{9 \cdot 2}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

Solve Using a Matrix with Cramer's Rule 7x+4y=5 , -x+2y=-2

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