3x-4y+2z=-1
2x+3y+5z=7
x+z=2

Answer

To solve the system of equations:

1. **Equations:**
\[ \begin{align*} (1) & \quad 3x - 4y + 2z = -1 \\ (2) & \quad 2x + 3y + 5z = 7 \\ (3) & \quad x + z = 2 \end{align*} \]

2. **Express \( z \) in terms of \( x \) from equation (3):**
\[ z = 2 - x \quad \text{(Equation 4)} \]

3. **Substitute \( z \) from equation (4) into equations (1) and (2):**

- Substitute into equation (1):
\[ 3x - 4y + 2(2 - x) = -1 \]
Simplifying this:
\[ 3x - 4y + 4 - 2x = -1 \]
\[ (3x - 2x) - 4y + 4 = -1 \]
\[ x - 4y + 4 = -1 \]
\[ x - 4y = -5 \quad \text{(Equation 5)} \]

- Substitute into equation (2):
\[ 2x + 3y + 5(2 - x) = 7 \]
Simplifying this:
\[ 2x + 3y + 10 - 5x = 7 \]
\[ (2x - 5x) + 3y + 10 = 7 \]
\[ -3x + 3y + 10 = 7 \]
\[ -3x + 3y = -3 \]
\[ -x + y = -1 \quad \text{(Equation 6)} \]

4. **Now we have two new equations (5) and (6):**
\[ \begin{align*} (5) & \quad x - 4y = -5 \\ (6) & \quad -x + y = -1 \end{align*} \]

5. **Add equations (5) and (6):**
\[ (x - 4y) + (-x + y) = -5 - 1 \]
This simplifies to:
\[ -3y = -6 \]
\[ y = 2 \]

6. **Substitute \( y = 2 \) back into equation (6) to find \( x \):**
\[ -x + 2 = -1 \]
\[ -x = -3 \]
\[ x = 3 \]

7. **Now substitute \( x = 3 \) back into equation (4) to find \( z \):**
\[ z = 2 - 3 = -1 \]

8. **Final solution:**
\[ \begin{align*} x & = 3 \\ y & = 2 \\ z & = -1 \end{align*} \]