`5z+4+4t+8+9=52`
`2z+4+w+8+9=34` In the correctly worked addition problems above, what is the value of `3z+4t-w`?

Answer

To solve the problem, we need to find the value of \(3z + 4t - w\) using the given equations:

1. **Write down the equations**:
\[ 5z + 4 + 4t + 8 + 9 = 52 \quad \text{(Equation 1)} \]
\[ 2z + 4 + w + 8 + 9 = 34 \quad \text{(Equation 2)} \]

2. **Simplify both equations**:
- For Equation 1:
\[ 5z + 4t + 4 + 8 + 9 = 52 \]
Combine the constants:
\[ 5z + 4t + 21 = 52 \]
Now, subtract 21 from both sides:
\[ 5z + 4t = 52 - 21 \]
\[ 5z + 4t = 31 \quad \text{(Equation 1 simplified)} \]

- For Equation 2:
\[ 2z + 4 + w + 8 + 9 = 34 \]
Combine the constants:
\[ 2z + w + 21 = 34 \]
Now, subtract 21 from both sides:
\[ 2z + w = 34 - 21 \]
\[ 2z + w = 13 \quad \text{(Equation 2 simplified)} \]

3. **Now we have two simplified equations**:
\[ 5z + 4t = 31 \quad \text{(1)} \]
\[ 2z + w = 13 \quad \text{(2)} \]

4. **Subtract Equation 2 from Equation 1**:
To eliminate \(w\), we can express \(w\) in terms of \(z\) from Equation 2:
\[ w = 13 - 2z \]
Now substitute \(w\) into the expression \(3z + 4t - w\):
\[ 3z + 4t - (13 - 2z) = 3z + 4t - 13 + 2z \]
Combine like terms:
\[ (3z + 2z) + 4t - 13 = 5z + 4t - 13 \]

5. **Substitute \(5z + 4t\) from Equation 1**:
From Equation 1, we know:
\[ 5z + 4t = 31 \]
Substitute this into the expression:
\[ 5z + 4t - 13 = 31 - 13 \]
\[ = 18 \]

6. **Final result**:
Thus, the value of \(3z + 4t - w\) is:
\[ \boxed{18} \]