Find the value of *Y* from the system of equations:

$$\begin{cases} X - 2Y + 2Z = 16 \ X - Y + Z = 9 \ 2X - 3Y - Z = 9 \end{cases}$$

Question

Find the value of *Y* from the system of equations:

$$\begin{cases} X - 2Y + 2Z = 16 \ X - Y + Z = 9 \ 2X - 3Y - Z = 9 \end{cases}$$

Accepted Answer

**Solution:**

1. Given the system:
   $$
   \begin{cases}
   X - 2Y + 2Z = 16 \quad (1) \
   X - Y + Z = 9 \quad (2) \
   2X - 3Y - Z = 9 \quad (3)
   \end{cases}
   $$

2. Subtract equation (2) from equation (1):
   $$
   (X - 2Y + 2Z) - (X - Y + Z) = 16 - 9
   $$
   Simplifying,
   $$
   -Y + Z = 7 \quad (4)
   $$

3. Multiply equation (2) by 2 and subtract it from equation (3):
   $$
   (2X - 3Y - Z) - 2(X - Y + Z) = 9 - 18
   $$
   Simplifying,
   $$
   (2X - 3Y - Z) - (2X - 2Y + 2Z) = -9
   $$
   $$
   -Y - 3Z = -9 \quad (5)
   $$

4. Subtract equation (5) from equation (4):
   $$
   (-Y + Z) - (-Y - 3Z) = 7 - (-9)
   $$
   $$
   4Z = 16
   $$
   $$
   Z = 4
   $$

5. Substitute $Z = 4$ into equation (4):
   $$
   -Y + 4 = 7
   $$
   $$
   -Y = 3
   $$
   $$
   Y = -3
   $$

6. Substitute $Y = -3$ and $Z = 4$ into equation (2) to find $X$:
   $$
   X - (-3) + 4 = 9
   $$
   $$
   X + 3 + 4 = 9
   $$
   $$
   X + 7 = 9
   $$
   $$
   X = 2
   $$

**Answer:** $Y = -3$

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