To solve the given system of linear equations, follow these steps:

Step 1: Write the system clearly

[ \begin{cases} x + y + z = 2 \quad (1) \ 2x - y + 3z = 1 \quad (2) \ 3x + 2y - z = 4 \quad (3) \end{cases} ]

Step 2: Eliminate (y) using equations (1) and (2)

Add (1) and (2):
[ (x+2x) + (y - y) + (z + 3z) = 2 + 1 \ 3x + 4z = 3 \quad (4) ]

Step 3: Eliminate (y) using equations (2) and (3)

Multiply (2) by 2: (4x - 2y + 6z = 2 \quad (5))
Add (5) and (3):
[ (4x+3x) + (-2y+2y) + (6z - z) = 2 + 4 \ 7x + 5z = 6 \quad (6) ]

Step 4: Solve for (x) and (z)

Multiply (4) by 7: (21x + 28z = 21 \quad (7))
Multiply (6) by 3: (21x + 15z = 18 \quad (8))
Subtract (8) from (7):
[ 13z = 3 \implies z = \frac{3}{13} ]
Substitute (z = \frac{3}{13}) into (4):
[ 3x + 4\left(\frac{3}{13}\right) = 3 \implies 3x = \frac{27}{13} \implies x = \frac{9}{13} ]

Step 5: Find (y)

Substitute (x = \frac{9}{13}) and (z = \frac{3}{13}) into (1):
[ \frac{9}{13} + y + \frac{3}{13} = 2 \implies y = \frac{14}{13} ]

Final Solution:
[ \boxed{x=\frac{9}{13}, y=\frac{14}{13}, z=\frac{3}{13}} ]

作者声明:本文包含人工智能生成内容。