Question

$$\begin{cases}x - y + z = 17\\-4x + y + 5z = -2\\2x + 3y + z = 8end{cases}$$

Explanation:

Step1: Write the system of equations

We have the system:

$$
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Step2: Eliminate \(y\) from the first two equations

Add the first and second equations:
\((x - y + z) + (-4x + y + 5z) = 17 + (-2)\)
Simplify:
\(x - 4x - y + y + z + 5z = 15\)
\(-3x + 6z = 15\)
Divide by -3:
\(x - 2z = -5\) (Equation 4)

Step3: Eliminate \(y\) from the second and third equations

Multiply the second equation by 3: \( -12x + 3y + 15z = -6\)
Subtract the third equation from this:
\((-12x + 3y + 15z) - (2x + 3y + z) = -6 - 8\)
Simplify:
\(-12x - 2x + 3y - 3y + 15z - z = -14\)
\(-14x + 14z = -14\)
Divide by -14:
\(x - z = 1\) (Equation 5)

Step4: Solve Equations 4 and 5

Equation 4: \(x - 2z = -5\)
Equation 5: \(x - z = 1\)
Subtract Equation 4 from Equation 5:
\((x - z) - (x - 2z) = 1 - (-5)\)
\(x - z - x + 2z = 6\)
\(z = 6\)

Step5: Find \(x\) using Equation 5

Substitute \(z = 6\) into \(x - z = 1\):
\(x - 6 = 1\)
\(x = 7\)

Step6: Find \(y\) using the first equation

Substitute \(x = 7\) and \(z = 6\) into \(x - y + z = 17\):
\(7 - y + 6 = 17\)
\(13 - y = 17\)
\(-y = 4\)
\(y = -4\)

Answer:

\(x = 7\), \(y = -4\), \(z = 6\)