`P=4x-z+3y`
`Q=-x+4z+3y`
Using the definitions above for P and Q. What is `2P-Q`?

Options

A. `7x-6z+3y`
B. `9x+2z+9y`
C. `9x-6z+3y`
D. `7x-6z+9y`

Answer

To solve the problem, we need to find the expression for \(2P - Q\) given the definitions of \(P\) and \(Q\).

1. **Identify the expressions for \(P\) and \(Q\)**:
\[ P = 4x - z + 3y \]
\[ Q = -x + 4z + 3y \]

2. **Calculate \(2P\)**:
\[ 2P = 2(4x - z + 3y) \]
Distributing the 2:
\[ 2P = 2 \cdot 4x + 2 \cdot (-z) + 2 \cdot 3y = 8x - 2z + 6y \]

3. **Substitute \(Q\) into the expression \(2P - Q\)**:
\[ 2P - Q = (8x - 2z + 6y) - (-x + 4z + 3y) \]

4. **Distribute the negative sign across \(Q\)**:
\[ 2P - Q = 8x - 2z + 6y + x - 4z - 3y \]

5. **Combine like terms**:
- Combine the \(x\) terms: \(8x + x = 9x\)
- Combine the \(z\) terms: \(-2z - 4z = -6z\)
- Combine the \(y\) terms: \(6y - 3y = 3y\)

Therefore, we have:
\[ 2P - Q = 9x - 6z + 3y \]

6. **Final Result**:
The expression for \(2P - Q\) is:
\[ \boxed{9x - 6z + 3y} \]