Solve the following equation for \( z \). \[ \sqrt{\frac{53+5 z}{7}}=2 \]

To solve the equation for \( z \), we start with the given equation: \[ \sqrt{\frac{53 + 5z}{7}} = 2 \] First, we square both sides to eliminate the square root: \[ \frac{53 + 5z}{7} = 2^2 \] This simplifies to: \[ \frac{53 + 5z}{7} = 4 \] Next, we multiply both sides by 7 to remove the fraction: \[ 53 + 5z = 4 \cdot 7 \] This simplifies to: \[ 53 + 5z = 28 \] Now, we isolate \( 5z \) by subtracting 53 from both sides: \[ 5z = 28 - 53 \] This results in: \[ 5z = -25 \] Next, we divide both sides by 5 to solve for \( z \): \[ z = \frac{-25}{5} \] This gives us: \[ z = -5 \] Thus, the solution for \( z \) is: \[ \boxed{-5} \]