1. **Stating the problem:** Solve the system of linear equations: $$\begin{cases} 2x + y - 2z = 1 \\ 3x - 2y + 4z = -2 \\ x + 3y + 2z = 4 \end{cases}$$ 2. **Formula and rules:** We will use the method of substitution or elimination to find values of $x$, $y$, and $z$ that satisfy all three equations simultaneously. 3. **Step 1: Express $x$ from the third equation:** $$x + 3y + 2z = 4 \implies x = 4 - 3y - 2z$$ 4. **Step 2: Substitute $x$ into the first and second equations:** First equation: $$2(4 - 3y - 2z) + y - 2z = 1$$ Simplify: $$8 - 6y - 4z + y - 2z = 1$$ $$8 - 5y - 6z = 1$$ Subtract 8 from both sides: $$-5y - 6z = 1 - 8$$ $$-5y - 6z = -7$$ Multiply both sides by $-1$: $$5y + 6z = 7$$ Second equation: $$3(4 - 3y - 2z) - 2y + 4z = -2$$ Simplify: $$12 - 9y - 6z - 2y + 4z = -2$$ $$12 - 11y - 2z = -2$$ Subtract 12 from both sides: $$-11y - 2z = -14$$ Multiply both sides by $-1$: $$11y + 2z = 14$$ 5. **Step 3: Solve the system of two equations with two variables:** $$\begin{cases} 5y + 6z = 7 \\ 11y + 2z = 14 \end{cases}$$ Multiply the first equation by 11 and the second by 5 to eliminate $y$: $$\begin{cases} 55y + 66z = 77 \\ 55y + 10z = 70 \end{cases}$$ Subtract the second from the first: $$55y + 66z - (55y + 10z) = 77 - 70$$ $$55y + 66z - 55y - 10z = 7$$ $$56z = 7$$ Divide both sides by 56: $$z = \frac{7}{56} = \frac{1}{8}$$ 6. **Step 4: Substitute $z = \frac{1}{8}$ into one of the two-variable equations:** Using $5y + 6z = 7$: $$5y + 6 \times \frac{1}{8} = 7$$ $$5y + \frac{6}{8} = 7$$ $$5y + \frac{3}{4} = 7$$ Subtract $\frac{3}{4}$ from both sides: $$5y = 7 - \frac{3}{4} = \frac{28}{4} - \frac{3}{4} = \frac{25}{4}$$ Divide both sides by 5: $$y = \frac{25}{4} \times \frac{1}{5} = \frac{25}{20} = \frac{5}{4}$$ 7. **Step 5: Substitute $y = \frac{5}{4}$ and $z = \frac{1}{8}$ into the expression for $x$:** $$x = 4 - 3y - 2z = 4 - 3 \times \frac{5}{4} - 2 \times \frac{1}{8}$$ Calculate: $$4 - \frac{15}{4} - \frac{2}{8} = 4 - \frac{15}{4} - \frac{1}{4} = 4 - \frac{16}{4} = 4 - 4 = 0$$ **Final solution:** $$\boxed{\left(x, y, z\right) = \left(0, \frac{5}{4}, \frac{1}{8}\right)}$$