Solve Using a Matrix with Cramer's Rule 8x-8y-8z=-48 , 2x+2y+4z=50 , -3x-8y-4z=-105

$8 x - 8 y - 8 z = - 48$ , $2 x + 2 y + 4 z = 50$ , $- 3 x - 8 y - 4 z = - 105$

Step 1

Represent the system of equations in matrix format.

$[\begin{array}{c} 8 & - 8 & - 8 \\ 2 & 2 & 4 \\ - 3 & - 8 & - 4 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} - 48 \\ 50 \\ - 105 \end{array}]$

Step 2

Find the determinant of $[\begin{array}{c} 8 & - 8 & - 8 \\ 2 & 2 & 4 \\ - 3 & - 8 & - 4 \end{array}]$ .

Set up the determinant by breaking it into smaller components.

$D = 8 | \begin{array}{c} 2 & 4 \\ - 8 & - 4 \end{array} | - 2 | \begin{array}{c} - 8 & - 8 \\ - 8 & - 4 \end{array} | - 3 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Find the determinant of $8 | \begin{array}{c} 2 & 4 \\ - 8 & - 4 \end{array} |$ .

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

$D = 8 ((2) (- 4) + 8 \cdot 4) - 2 | \begin{array}{c} - 8 & - 8 \\ - 8 & - 4 \end{array} | - 3 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $- 4$ .

$D = 8 (- 8 + 8 \cdot 4) - 2 | \begin{array}{c} - 8 & - 8 \\ - 8 & - 4 \end{array} | - 3 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Multiply $8$ by $4$ .

$D = 8 (- 8 + 32) - 2 | \begin{array}{c} - 8 & - 8 \\ - 8 & - 4 \end{array} | - 3 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Simplify the expression.

Add $- 8$ and $32$ .

$D = 8 \cdot 24 - 2 | \begin{array}{c} - 8 & - 8 \\ - 8 & - 4 \end{array} | - 3 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Multiply $8$ by $24$ .

$D = 192 - 2 | \begin{array}{c} - 8 & - 8 \\ - 8 & - 4 \end{array} | - 3 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Find the determinant of $- 2 | \begin{array}{c} - 8 & - 8 \\ - 8 & - 4 \end{array} |$ .

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

$D = 192 - 2 ((- 8) (- 4) + 8 \cdot - 8) - 3 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $- 8$ by $- 4$ .

$D = 192 - 2 (32 + 8 \cdot - 8) - 3 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Multiply $8$ by $- 8$ .

$D = 192 - 2 (32 - 64) - 3 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Simplify the expression.

Subtract $64$ from $32$ .

$D = 192 - 2 \cdot - 32 - 3 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Multiply $- 2$ by $- 32$ .

$D = 192 + 64 - 3 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Find the determinant of $- 3 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$ .

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

$D = 192 + 64 - 3 ((- 8) (4) - 2 \cdot - 8)$

Simplify the determinant.

Simplify each term.

Multiply $- 8$ by $4$ .

$D = 192 + 64 - 3 (- 32 - 2 \cdot - 8)$

Multiply $- 2$ by $- 8$ .

$D = 192 + 64 - 3 (- 32 + 16)$

Simplify the expression.

Add $- 32$ and $16$ .

$D = 192 + 64 - 3 \cdot - 16$

Multiply $- 3$ by $- 16$ .

$D = 192 + 64 + 48$

Add $192$ and $64$ .

$D = 256 + 48$

Add $256$ and $48$ .

$D = 304$

Step 3

Find the determinant of $[\begin{array}{c} - 48 & - 8 & - 8 \\ 50 & 2 & 4 \\ - 105 & - 8 & - 4 \end{array}]$ .

Set up the determinant by breaking it into smaller components.

$D_{x} = - 48 | \begin{array}{c} 2 & 4 \\ - 8 & - 4 \end{array} | - 50 | \begin{array}{c} - 8 & - 8 \\ - 8 & - 4 \end{array} | - 105 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Find the determinant of $- 48 | \begin{array}{c} 2 & 4 \\ - 8 & - 4 \end{array} |$ .

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

$D_{x} = - 48 ((2) (- 4) + 8 \cdot 4) - 50 | \begin{array}{c} - 8 & - 8 \\ - 8 & - 4 \end{array} | - 105 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $- 4$ .

$D_{x} = - 48 (- 8 + 8 \cdot 4) - 50 | \begin{array}{c} - 8 & - 8 \\ - 8 & - 4 \end{array} | - 105 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Multiply $8$ by $4$ .

$D_{x} = - 48 (- 8 + 32) - 50 | \begin{array}{c} - 8 & - 8 \\ - 8 & - 4 \end{array} | - 105 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Simplify the expression.

Add $- 8$ and $32$ .

$D_{x} = - 48 \cdot 24 - 50 | \begin{array}{c} - 8 & - 8 \\ - 8 & - 4 \end{array} | - 105 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Multiply $- 48$ by $24$ .

$D_{x} = - 1152 - 50 | \begin{array}{c} - 8 & - 8 \\ - 8 & - 4 \end{array} | - 105 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Find the determinant of $- 50 | \begin{array}{c} - 8 & - 8 \\ - 8 & - 4 \end{array} |$ .

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

$D_{x} = - 1152 - 50 ((- 8) (- 4) + 8 \cdot - 8) - 105 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $- 8$ by $- 4$ .

$D_{x} = - 1152 - 50 (32 + 8 \cdot - 8) - 105 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Multiply $8$ by $- 8$ .

$D_{x} = - 1152 - 50 (32 - 64) - 105 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Simplify the expression.

Subtract $64$ from $32$ .

$D_{x} = - 1152 - 50 \cdot - 32 - 105 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Multiply $- 50$ by $- 32$ .

$D_{x} = - 1152 + 1600 - 105 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$

Find the determinant of $- 105 | \begin{array}{c} - 8 & - 8 \\ 2 & 4 \end{array} |$ .

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

$D_{x} = - 1152 + 1600 - 105 ((- 8) (4) - 2 \cdot - 8)$

Simplify the determinant.

Simplify each term.

Multiply $- 8$ by $4$ .

$D_{x} = - 1152 + 1600 - 105 (- 32 - 2 \cdot - 8)$

Multiply $- 2$ by $- 8$ .

$D_{x} = - 1152 + 1600 - 105 (- 32 + 16)$

Simplify the expression.

Add $- 32$ and $16$ .

$D_{x} = - 1152 + 1600 - 105 \cdot - 16$

Multiply $- 105$ by $- 16$ .

$D_{x} = - 1152 + 1600 + 1680$

Add $- 1152$ and $1600$ .

$D_{x} = 448 + 1680$

Add $448$ and $1680$ .

$D_{x} = 2128$

Step 4

Find the determinant of $[\begin{array}{c} 8 & - 48 & - 8 \\ 2 & 50 & 4 \\ - 3 & - 105 & - 4 \end{array}]$ .

Set up the determinant by breaking it into smaller components.

$D_{y} = 8 | \begin{array}{c} 50 & 4 \\ - 105 & - 4 \end{array} | - 2 | \begin{array}{c} - 48 & - 8 \\ - 105 & - 4 \end{array} | - 3 | \begin{array}{c} - 48 & - 8 \\ 50 & 4 \end{array} |$

Find the determinant of $8 | \begin{array}{c} 50 & 4 \\ - 105 & - 4 \end{array} |$ .

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

$D_{y} = 8 ((50) (- 4) + 105 \cdot 4) - 2 | \begin{array}{c} - 48 & - 8 \\ - 105 & - 4 \end{array} | - 3 | \begin{array}{c} - 48 & - 8 \\ 50 & 4 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $50$ by $- 4$ .

$D_{y} = 8 (- 200 + 105 \cdot 4) - 2 | \begin{array}{c} - 48 & - 8 \\ - 105 & - 4 \end{array} | - 3 | \begin{array}{c} - 48 & - 8 \\ 50 & 4 \end{array} |$

Multiply $105$ by $4$ .

$D_{y} = 8 (- 200 + 420) - 2 | \begin{array}{c} - 48 & - 8 \\ - 105 & - 4 \end{array} | - 3 | \begin{array}{c} - 48 & - 8 \\ 50 & 4 \end{array} |$

Simplify the expression.

Add $- 200$ and $420$ .

$D_{y} = 8 \cdot 220 - 2 | \begin{array}{c} - 48 & - 8 \\ - 105 & - 4 \end{array} | - 3 | \begin{array}{c} - 48 & - 8 \\ 50 & 4 \end{array} |$

Multiply $8$ by $220$ .

$D_{y} = 1760 - 2 | \begin{array}{c} - 48 & - 8 \\ - 105 & - 4 \end{array} | - 3 | \begin{array}{c} - 48 & - 8 \\ 50 & 4 \end{array} |$

Find the determinant of $- 2 | \begin{array}{c} - 48 & - 8 \\ - 105 & - 4 \end{array} |$ .

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

$D_{y} = 1760 - 2 ((- 48) (- 4) + 105 \cdot - 8) - 3 | \begin{array}{c} - 48 & - 8 \\ 50 & 4 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $- 48$ by $- 4$ .

$D_{y} = 1760 - 2 (192 + 105 \cdot - 8) - 3 | \begin{array}{c} - 48 & - 8 \\ 50 & 4 \end{array} |$

Multiply $105$ by $- 8$ .

$D_{y} = 1760 - 2 (192 - 840) - 3 | \begin{array}{c} - 48 & - 8 \\ 50 & 4 \end{array} |$

Simplify the expression.

Subtract $840$ from $192$ .

$D_{y} = 1760 - 2 \cdot - 648 - 3 | \begin{array}{c} - 48 & - 8 \\ 50 & 4 \end{array} |$

Multiply $- 2$ by $- 648$ .

$D_{y} = 1760 + 1296 - 3 | \begin{array}{c} - 48 & - 8 \\ 50 & 4 \end{array} |$

Find the determinant of $- 3 | \begin{array}{c} - 48 & - 8 \\ 50 & 4 \end{array} |$ .

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

$D_{y} = 1760 + 1296 - 3 ((- 48) (4) - 50 \cdot - 8)$

Simplify the determinant.

Simplify each term.

Multiply $- 48$ by $4$ .

$D_{y} = 1760 + 1296 - 3 (- 192 - 50 \cdot - 8)$

Multiply $- 50$ by $- 8$ .

$D_{y} = 1760 + 1296 - 3 (- 192 + 400)$

Simplify the expression.

Add $- 192$ and $400$ .

$D_{y} = 1760 + 1296 - 3 \cdot 208$

Multiply $- 3$ by $208$ .

$D_{y} = 1760 + 1296 - 624$

Add $1760$ and $1296$ .

$D_{y} = 3056 - 624$

Subtract $624$ from $3056$ .

$D_{y} = 2432$

Step 5

Find the determinant of $[\begin{array}{c} 8 & - 8 & - 48 \\ 2 & 2 & 50 \\ - 3 & - 8 & - 105 \end{array}]$ .

Set up the determinant by breaking it into smaller components.

$D_{z} = 8 | \begin{array}{c} 2 & 50 \\ - 8 & - 105 \end{array} | - 2 | \begin{array}{c} - 8 & - 48 \\ - 8 & - 105 \end{array} | - 3 | \begin{array}{c} - 8 & - 48 \\ 2 & 50 \end{array} |$

Find the determinant of $8 | \begin{array}{c} 2 & 50 \\ - 8 & - 105 \end{array} |$ .

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

$D_{z} = 8 ((2) (- 105) + 8 \cdot 50) - 2 | \begin{array}{c} - 8 & - 48 \\ - 8 & - 105 \end{array} | - 3 | \begin{array}{c} - 8 & - 48 \\ 2 & 50 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $- 105$ .

$D_{z} = 8 (- 210 + 8 \cdot 50) - 2 | \begin{array}{c} - 8 & - 48 \\ - 8 & - 105 \end{array} | - 3 | \begin{array}{c} - 8 & - 48 \\ 2 & 50 \end{array} |$

Multiply $8$ by $50$ .

$D_{z} = 8 (- 210 + 400) - 2 | \begin{array}{c} - 8 & - 48 \\ - 8 & - 105 \end{array} | - 3 | \begin{array}{c} - 8 & - 48 \\ 2 & 50 \end{array} |$

Simplify the expression.

Add $- 210$ and $400$ .

$D_{z} = 8 \cdot 190 - 2 | \begin{array}{c} - 8 & - 48 \\ - 8 & - 105 \end{array} | - 3 | \begin{array}{c} - 8 & - 48 \\ 2 & 50 \end{array} |$

Multiply $8$ by $190$ .

$D_{z} = 1520 - 2 | \begin{array}{c} - 8 & - 48 \\ - 8 & - 105 \end{array} | - 3 | \begin{array}{c} - 8 & - 48 \\ 2 & 50 \end{array} |$

Find the determinant of $- 2 | \begin{array}{c} - 8 & - 48 \\ - 8 & - 105 \end{array} |$ .

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

$D_{z} = 1520 - 2 ((- 8) (- 105) + 8 \cdot - 48) - 3 | \begin{array}{c} - 8 & - 48 \\ 2 & 50 \end{array} |$

Simplify the determinant.

Simplify each term.

Multiply $- 8$ by $- 105$ .

$D_{z} = 1520 - 2 (840 + 8 \cdot - 48) - 3 | \begin{array}{c} - 8 & - 48 \\ 2 & 50 \end{array} |$

Multiply $8$ by $- 48$ .

$D_{z} = 1520 - 2 (840 - 384) - 3 | \begin{array}{c} - 8 & - 48 \\ 2 & 50 \end{array} |$

Simplify the expression.

Subtract $384$ from $840$ .

$D_{z} = 1520 - 2 \cdot 456 - 3 | \begin{array}{c} - 8 & - 48 \\ 2 & 50 \end{array} |$

Multiply $- 2$ by $456$ .

$D_{z} = 1520 - 912 - 3 | \begin{array}{c} - 8 & - 48 \\ 2 & 50 \end{array} |$

Find the determinant of $- 3 | \begin{array}{c} - 8 & - 48 \\ 2 & 50 \end{array} |$ .

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

$D_{z} = 1520 - 912 - 3 ((- 8) (50) - 2 \cdot - 48)$

Simplify the determinant.

Simplify each term.

Multiply $- 8$ by $50$ .

$D_{z} = 1520 - 912 - 3 (- 400 - 2 \cdot - 48)$

Multiply $- 2$ by $- 48$ .

$D_{z} = 1520 - 912 - 3 (- 400 + 96)$

Simplify the expression.

Add $- 400$ and $96$ .

$D_{z} = 1520 - 912 - 3 \cdot - 304$

Multiply $- 3$ by $- 304$ .

$D_{z} = 1520 - 912 + 912$

Subtract $912$ from $1520$ .

$D_{z} = 608 + 912$

Add $608$ and $912$ .

$D_{z} = 1520$

Step 6

Find the value of $x$ by Cramer\'s Rule, which states that $x = \frac{D_{x}}{D}$ . In this case, $x = \frac{2128}{304}$ .

Remove parentheses.

$x = \frac{2128}{304}$

Divide $2128$ by $304$ .

$x = 7$

Step 7

Find the value of $y$ by Cramer\'s Rule, which states that $y = \frac{D_{y}}{D}$ . In this case, $y = \frac{2432}{304}$ .

Remove parentheses.

$y = \frac{2432}{304}$

Divide $2432$ by $304$ .

$y = 8$

Step 8

Find the value of $z$ by Cramer\'s Rule, which states that $z = \frac{D_{z}}{D}$ . In this case, $z = \frac{1520}{304}$ .

Remove parentheses.

$z = \frac{1520}{304}$

Divide $1520$ by $304$ .

$z = 5$

Step 9

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

$x = 7, y = 8, z = 5$

Solve Using a Matrix with Cramer's Rule 8x-8y-8z=-48 , 2x+2y+4z=50 , -3x-8y-4z=-105

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