Solve the system x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2

Problem: Solve the system of linear equations: $x + y + z = 6$, $2x - y + z = 3$, $x + 2y - z = 2$.

Steps

1. Label the equations
   $$\text{(1) } x + y + z = 6, \quad \text{(2) } 2x - y + z = 3, \quad \text{(3) } x + 2y - z = 2$$
2. Add equations (1) and (2) to eliminate $y$
   $$3x + 2z = 9$$
3. Add equations (2) and (3) to eliminate $y$
   $$3x = 5$$
4. Solve for $x$
   $$x = \frac{5}{3}$$
5. Substitute $x = \frac{5}{3}$ into $3x + 2z = 9$
   $$3\left(\frac{5}{3}\right) + 2z = 9$$
6. Simplify and solve for $z$
   $$5 + 2z = 9 \implies 2z = 4 \implies z = 2$$
7. Substitute $x = \frac{5}{3}$ and $z = 2$ into equation (1)
   $$\frac{5}{3} + y + 2 = 6$$
8. Solve for $y$
   $$y + \frac{11}{3} = 6 \implies y = 6 - \frac{11}{3} = \frac{7}{3}$$