To solve the system of equations for \( z \) and \( y \), we can start with the two equations given:
- \( y = 9 + 3z \)
- \( y = 58 - 4z \)
Since both expressions equal \( y \), we can set them equal to each other:
\[ 9 + 3z = 58 - 4z \]
Next, we will solve for \( z \). First, we can add \( 4z \) to both sides:
\[ 9 + 3z + 4z = 58 \]
This simplifies to:
\[ 9 + 7z = 58 \]
Now, subtract 9 from both sides:
\[ 7z = 49 \]
Next, divide both sides by 7:
\[ z = 7 \]
Now that we have the value for \( z \), we can substitute it back into either of the original equations to find the value of \( y \). We'll use the first equation:
\[ y = 9 + 3z \]
Substituting \( z = 7 \):
\[ y = 9 + 3(7) \]
Calculating it:
\[ y = 9 + 21 \] \[ y = 30 \]
So, the solutions are: \[ z = 7 \] \[ y = 30 \]