Question

1. -2x + 5y + 3z = -25

-4x - 3y - 8z = -39
6x + 8y - 5z = 14

Explanation:

Step1: Eliminate x from first two equations

Multiply the first equation \(-2x + 5y + 3z = -25\) by 2: \( -4x + 10y + 6z = -50 \).
Subtract the second equation \(-4x - 3y - 8z = -39\) from this new equation:
\((-4x + 10y + 6z) - (-4x - 3y - 8z) = -50 - (-39)\)
Simplify: \(13y + 14z = -11\) (Equation 4)

Step2: Eliminate x from first and third equations

Multiply the first equation \(-2x + 5y + 3z = -25\) by 3: \( -6x + 15y + 9z = -75 \).
Add to the third equation \(6x + 8y - 5z = 14\):
\((-6x + 15y + 9z) + (6x + 8y - 5z) = -75 + 14\)
Simplify: \(23y + 4z = -61\) (Equation 5)

Step3: Solve Equations 4 and 5 for y and z

Multiply Equation 5 by 7: \(161y + 28z = -427\) (Equation 6)
Multiply Equation 4 by 2: \(26y + 28z = -22\) (Equation 7)
Subtract Equation 7 from Equation 6:
\((161y + 28z) - (26y + 28z) = -427 - (-22)\)
Simplify: \(135y = -405\) → \(y = -3\)

Step4: Substitute y = -3 into Equation 4

\(13(-3) + 14z = -11\) → \(-39 + 14z = -11\) → \(14z = 28\) → \(z = 2\)

Step5: Substitute y = -3, z = 2 into first equation

\(-2x + 5(-3) + 3(2) = -25\) → \(-2x - 15 + 6 = -25\) → \(-2x - 9 = -25\) → \(-2x = -16\) → \(x = 8\)

Answer:

\(x = 8\), \(y = -3\), \(z = 2\)