Question

solve the system of equations using cramer's rule -x+y-3z=-4 3x-2y+8z=14 2x-2y+5z=7

Answer 1

System of Equations
$-1x + 1y - 3z = -4 \\3x - 2y + 8z = 14 \\2x - 2y + 5z = 7$

Coefficient Matrix's Determinant
$D = \left[\begin{array}{ccc}-1&1&-3\\3&-2&8\\2&-2&5\end{array}\right]$

Answer Column
$\left[\begin{array}{ccc}-4\\14\\7\end{array}\right]$

Dx: Coefficient Determinant with Answer-Column values in X-Column
$D_{x} = \left[\begin{array}{ccc}-4&1&-3\\14&-2&8\\7&-2&5\end{array}\right]$

Dy: Coefficient Determinant with Answer-Column Values in Y-Column
$D_{y} = \left[\begin{array}{ccc}-1&-4&-3\\3&14&8\\2&7&5\end{array}\right]$

Dz: Coefficient Determinant with Answer-Column Values in Z-Column
$D_{z} = \left[\begin{array}{ccc}-1&1&-4\\3&-2&14\\2&-2&7\end{array}\right]$

Evaluating each Determinant
$D= \left[\begin{array}{ccc}-1&1&-3\\3&-2&8\\2&-2&5\end{array}\right] \\D = (-1 * (-2) * 5) + (1 * 8 * 2) + (-3 * 3 * (-2)) - (2 * (-2) * (-3)) - (-2 * 8 * (-1)) - (5 * 3 * 1) \\D = (10) + (16) + (18) - (12) - (16) - (15) \\D = 10 + 16 + 18 - 12 - 16 - 15 \\D = 26 + 18 - 12 - 16 - 15 \\D = 44 - 12 - 16 - 15 \\D = 32 - 16 - 15 \\D = 16 - 15 \\D = 1$

$D_{x} = \left[\begin{array}{ccc}-4&1&-3\\14&-2&8\\7&-2&5\end{array}\right] \\D_{x} = (-4 * (-2) * 5) + (1 * 8 * 7) + (-3 * 14 * (-2)) - (7 * (-2) * (-3)) - (-2 * 8 * (-4)) - (5 * 14 * 1)) \\D_{x} = (40) + (56) + (84) - (42) - (64) - (70) \\D_{x} = 40 + 56 + 84 - 42 - 64 - 70 \\D_{x} = 96 + 84 - 42 - 64 - 70 \\D_{x} = 180 - 42 - 64 - 70 \\D_{x} = 138 - 64 - 70 \\D_{x} = 74 - 70 \\D_{x} = 4$

$D_{y} = \left[\begin{array}{ccc}-1&-4&-3\\3&14&8\\2&7&5\end{array}\right] \\D_{y} = (-1 * 14 * 5) + (-4 * 8 * 2) + (-3 * 3 * 7) - (2 * 14 * (-3)) - (7 * 8 * (-1)) * (5 * 3 * (-4)) \\D_{y} = (-70)+ (-64) + (-63) - (-84) - (-56) - (-60) \\D_{y} = -70 - 64 - 63 + 84 + 56 + 60 \\D_{y} = -134 - 63 + 84 + 56 + 60 \\D_{y} = -197 + 84 + 56 + 60 \\D_{y} = -113 + 56 + 60 \\D_{y} = -57 + 60 \\D_{y} = 3$

$D_{z} = \left[\begin{array}{ccc}-1&1&-4\\3&-2&14\\2&-2&7\end{array}\right] \\D_{z} = (-1 * (-2) * 7) + (1 * 14 * 2) + (-4 * 3 * (-2)) - (2 * (-2) * (-4)) - (-2 * 14 * (-1)) - (7 * 3 * 1) \\D_{z} = (14) + (28) + (24) - (16) - (28) - (21) \\D_{z} = 14 + 28 + 24 - 16 - 28 - 24 \\D_{z} = 42 + 24 - 16 - 28 - 21 \\D_{z} = 66 - 16 - 28 - 21 \\D_{z} = 50 - 28 - 21 \\D_{z} = 22 - 21 \\D_{z} = 1$

$x = \frac{D_{x}}{D} = \frac{4}{1} = 4 \\y = \frac{D_{y}}{D} = \frac{3}{1} = 3 \\z = \frac{D_{x}}{D} = \frac{1}{1} = 1 \\(x, y, z) = (4, 3, 1)$