Question

$$\begin{cases}x_1 - x_2 + 2x_3 = 3,\\4x_1 - 3x_2 + 2x_3 = 2,\\3x_1 - 2x_2 + x_3 = 1end{cases}$$

Explanation:

Step1: Label the equations

Let's label the equations as follows:

• Equation (1): \(x_1 - x_2 + 2x_3 = 3\)
• Equation (2): \(4x_1 - 3x_2 + 2x_3 = 2\)
• Equation (3): \(3x_1 - 2x_2 + x_3 = 1\)

Step2: Eliminate \(x_3\) from Equation (1) and Equation (2)

Subtract Equation (1) from Equation (2):

$$
LATEXBLOCK0
$$

Step3: Eliminate \(x_3\) from Equation (1) and Equation (3)

Multiply Equation (3) by 2: \(6x_1 - 4x_2 + 2x_3 = 2\) (Equation 5)
Subtract Equation (1) from Equation (5):

$$
LATEXBLOCK1
$$

Step4: Solve the system of Equation (4) and Equation (6)

Multiply Equation (4) by 3: \(9x_1 - 6x_2 = -3\) (Equation 7)
Multiply Equation (6) by 2: \(10x_1 - 6x_2 = -2\) (Equation 8)
Subtract Equation (7) from Equation (8):

$$
LATEXBLOCK2
$$

Step5: Substitute \(x_1 = 1\) into Equation (4)

$$
LATEXBLOCK3
$$

Step6: Substitute \(x_1 = 1\) and \(x_2 = 2\) into Equation (1)

$$
LATEXBLOCK4
$$

Answer:

\(x_1 = 1\), \(x_2 = 2\), \(x_3 = 2\)