Solve Using a Matrix with Cramer's Rule 11x+3y=1 , 5x+2y=4

$11 x + 3 y = 1$ , $5 x + 2 y = 4$

Represent the system of equations in matrix format.

$[\begin{array}{c} 11 & 3 \\ 5 & 2 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 1 \\ 4 \end{array}]$

Find the determinant of $[\begin{array}{c} 11 & 3 \\ 5 & 2 \end{array}]$ .

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

$determinant [\begin{array}{c} 11 & 3 \\ 5 & 2 \end{array}] = | \begin{array}{c} 11 & 3 \\ 5 & 2 \end{array} |$

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

$D = (11) (2) - 5 \cdot 3$

Simplify the determinant.

Simplify each term.

Multiply $11$ by $2$ .

$D = 22 - 5 \cdot 3$

Multiply $- 5$ by $3$ .

$D = 22 - 15$

Subtract $15$ from $22$ .

$D = 7$

Find the determinant of $[\begin{array}{c} 1 & 3 \\ 4 & 2 \end{array}]$ .

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

$determinant [\begin{array}{c} 1 & 3 \\ 4 & 2 \end{array}] = | \begin{array}{c} 1 & 3 \\ 4 & 2 \end{array} |$

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

$D_{x} = (1) (2) - 4 \cdot 3$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $1$ .

$D_{x} = 2 - 4 \cdot 3$

Multiply $- 4$ by $3$ .

$D_{x} = 2 - 12$

Subtract $12$ from $2$ .

$D_{x} = - 10$

Find the determinant of $[\begin{array}{c} 11 & 1 \\ 5 & 4 \end{array}]$ .

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

$determinant [\begin{array}{c} 11 & 1 \\ 5 & 4 \end{array}] = | \begin{array}{c} 11 & 1 \\ 5 & 4 \end{array} |$

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

$D_{y} = (11) (4) - 5 \cdot 1$

Simplify the determinant.

Simplify each term.

Multiply $11$ by $4$ .

$D_{y} = 44 - 5 \cdot 1$

Multiply $- 5$ by $1$ .

$D_{y} = 44 - 5$

Subtract $5$ from $44$ .

$D_{y} = 39$

Find the value of $x$ by Cramer\'s Rule, which states that $x = \frac{D_{x}}{D}$ . In this case, $x = \frac{- 10}{7}$ .

Remove parentheses.

$x = \frac{- 10}{7}$

Move the negative in front of the fraction.

$x = - \frac{10}{7}$

Remove parentheses.

$y = \frac{39}{7}$

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

$x = - \frac{10}{7}, y = \frac{39}{7}$

Solve Using a Matrix with Cramer's Rule 11x+3y=1 , 5x+2y=4

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