Question

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$$\begin{cases} -x_1 + 2x_2 = 3, -3x_1 + 2x_2 - x_3 = 4, 2x_1 - x_2 + x_3 = -2, \\end{cases}$$

Explanation:

Step1: Label the equations

Let the equations be:
Equation (1): \(-x_1 + 2x_2 = 3\)
Equation (2): \(-3x_1 + 2x_2 - x_3 = 4\)
Equation (3): \(2x_1 - x_2 + x_3 = -2\)

Step2: Eliminate \(x_3\) by adding Equation (2) and Equation (3)

Adding Equation (2) and Equation (3):

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Let's call this Equation (4): \(-x_1 + x_2 = 2\)

Step3: Subtract Equation (4) from Equation (1) to find \(x_2\)

Equation (1): \(-x_1 + 2x_2 = 3\)
Equation (4): \(-x_1 + x_2 = 2\)
Subtract Equation (4) from Equation (1):

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Step4: Substitute \(x_2 = 1\) into Equation (1) to find \(x_1\)

Substitute \(x_2 = 1\) into Equation (1): \(-x_1 + 2(1) = 3\)
Simplify: \(-x_1 + 2 = 3\)
Subtract 2 from both sides: \(-x_1 = 3 - 2 = 1\)
Multiply both sides by -1: \(x_1 = -1\)

Step5: Substitute \(x_1 = -1\) and \(x_2 = 1\) into Equation (3) to find \(x_3\)

Substitute \(x_1 = -1\) and \(x_2 = 1\) into Equation (3): \(2(-1) - 1 + x_3 = -2\)
Simplify: \(-2 - 1 + x_3 = -2\)
\(-3 + x_3 = -2\)
Add 3 to both sides: \(x_3 = -2 + 3 = 1\)

Answer:

\(x_1 = -1\), \(x_2 = 1\), \(x_3 = 1\)