Question

1. $6x - 5y + 2z = -49$

$-5x - 3y - 8z = -22$
$-3x + 8y - 5z = 55$

Explanation:

Step1: Eliminate one variable (e.g., \(x\))

Multiply the first equation \(6x - 5y + 2z = -49\) by \(5\): \(30x - 25y + 10z = -245\)
Multiply the second equation \(-5x - 3y - 8z = -22\) by \(6\): \(-30x - 18y - 48z = -132\)
Add these two new equations:
\((30x - 25y + 10z) + (-30x - 18y - 48z) = -245 + (-132)\)
Simplify: \(-43y - 38z = -377\) --- Equation (4)

Multiply the first equation \(6x - 5y + 2z = -49\) by \(3\): \(18x - 15y + 6z = -147\)
Multiply the third equation \(-3x + 8y - 5z = 55\) by \(6\): \(-18x + 48y - 30z = 330\)
Add these two new equations:
\((18x - 15y + 6z) + (-18x + 48y - 30z) = -147 + 330\)
Simplify: \(33y - 24z = 183\) --- Equation (5)

Step2: Solve for \(y\) and \(z\)

Multiply Equation (4) by \(33\): \(-1419y - 1254z = -12441\)
Multiply Equation (5) by \(43\): \(1419y - 1032z = 7869\)
Add these two equations:
\((-1419y - 1254z) + (1419y - 1032z) = -12441 + 7869\)
Simplify: \(-2286z = -4572\)
Solve for \(z\): \(z = \frac{-4572}{-2286} = 2\)

Substitute \(z = 2\) into Equation (4): \(-43y - 38(2) = -377\)
\(-43y - 76 = -377\)
\(-43y = -377 + 76 = -301\)
\(y = \frac{-301}{-43} = 7\)

Step3: Solve for \(x\)

Substitute \(y = 7\) and \(z = 2\) into the first equation \(6x - 5(7) + 2(2) = -49\)
\(6x - 35 + 4 = -49\)
\(6x - 31 = -49\)
\(6x = -49 + 31 = -18\)
\(x = \frac{-18}{6} = -3\)

Answer:

\(x = -3\), \(y = 7\), \(z = 2\)