Solve Using a Matrix with Cramer's Rule 4x+5y=2 5x-3y=21

$4 x + 5 y = 2$ $5 x - 3 y = 21$

Step 1

Represent the system of equations in matrix format.

$[\begin{array}{c} 4 & 5 \\ 5 & - 3 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 2 \\ 21 \end{array}]$

Step 2

Find the determinant of $[\begin{array}{c} 4 & 5 \\ 5 & - 3 \end{array}]$ .

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

$determinant [\begin{array}{c} 4 & 5 \\ 5 & - 3 \end{array}] = | \begin{array}{c} 4 & 5 \\ 5 & - 3 \end{array} |$

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

$D = (4) (- 3) - 5 \cdot 5$

Simplify the determinant.

Simplify each term.

Multiply $4$ by $- 3$ .

$D = - 12 - 5 \cdot 5$

Multiply $- 5$ by $5$ .

$D = - 12 - 25$

Subtract $25$ from $- 12$ .

$D = - 37$

Step 3

Find the determinant of $[\begin{array}{c} 2 & 5 \\ 21 & - 3 \end{array}]$ .

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

$determinant [\begin{array}{c} 2 & 5 \\ 21 & - 3 \end{array}] = | \begin{array}{c} 2 & 5 \\ 21 & - 3 \end{array} |$

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

$D_{x} = (2) (- 3) - 21 \cdot 5$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $- 3$ .

$D_{x} = - 6 - 21 \cdot 5$

Multiply $- 21$ by $5$ .

$D_{x} = - 6 - 105$

Subtract $105$ from $- 6$ .

$D_{x} = - 111$

Step 4

Find the determinant of $[\begin{array}{c} 4 & 2 \\ 5 & 21 \end{array}]$ .

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

$determinant [\begin{array}{c} 4 & 2 \\ 5 & 21 \end{array}] = | \begin{array}{c} 4 & 2 \\ 5 & 21 \end{array} |$

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

$D_{y} = (4) (21) - 5 \cdot 2$

Simplify the determinant.

Simplify each term.

Multiply $4$ by $21$ .

$D_{y} = 84 - 5 \cdot 2$

Multiply $- 5$ by $2$ .

$D_{y} = 84 - 10$

Subtract $10$ from $84$ .

$D_{y} = 74$

Step 5

Find the value of $x$ by Cramer\'s Rule, which states that $x = \frac{D_{x}}{D}$ . In this case, $x = \frac{- 111}{- 37}$ .

Remove parentheses.

$x = \frac{- 111}{- 37}$

Divide $- 111$ by $- 37$ .

$x = 3$

Step 6

Find the value of $y$ by Cramer\'s Rule, which states that $y = \frac{D_{y}}{D}$ . In this case, $y = \frac{74}{- 37}$ .

Remove parentheses.

$y = \frac{74}{- 37}$

Divide $74$ by $- 37$ .

$y = - 2$

Step 7

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

$x = 3, y = - 2$

Step 8

Solve Using a Matrix with Cramer's Rule 4x+5y=2 5x-3y=21

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