Question

solve the system of equations:
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$$\begin{cases} 2x - 5y + z = 16 \\\\ -x + y - z = -4 \\\\ x - 3y = 9 \\end{cases}$$

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\\(x = \\)
\\(y = \\)
\\(z = \\)

Explanation:

Step1: Write down the system of equations

$$
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Step2: Eliminate \(z\) by adding Equation 1 and Equation 2

$$ (2x - 5y + z) + (-x + y - z) = 16 + (-4) \implies x - 4y = 12 \quad \text{(Equation 4)} $$

Step3: Solve for \(y\) using Equation 3 and Equation 4

$$ (x - 4y) - (x - 3y) = 12 - 9 \implies -y = 3 \implies y = -2 $$

Step4: Substitute \(y\) into Equation 3 to find \(x\)

$$ x - 3(-2) = 9 \implies x + 6 = 9 \implies x = 3 $$

Step5: Substitute \(x\) and \(y\) into Equation 2 to find \(z\)

$$ -(3) + (-2) - z = -4 \implies -5 - z = -4 \implies z = -1 \times (5 - 4) \implies z = -4 $$

Answer:

\(x = 3\)
\(y = -2\)
\(z = -4\)