The equation you provided is:

$\tan(45^\circ) = \frac{\frac{9 - y}{5} - \frac{6}{11}}{1 + \left(\frac{9 - y}{5}\right)\left(\frac{6}{11}\right)}$

Since $\tan(45^\circ) = 1$, we can set the left side equal to 1:

$1 = \frac{\frac{9 - y}{5} - \frac{6}{11}}{1 + \left(\frac{9 - y}{5}\right)\left(\frac{6}{11}\right)}$

Next, let's solve for $y$.

Step 1: Multiply both sides of the equation by the denominator on the right-hand side to clear the fraction:

$1 \times \left(1 + \left(\frac{9 - y}{5}\right)\left(\frac{6}{11}\right)\right) = \frac{9 - y}{5} - \frac{6}{11}$

This simplifies to: $1 + \left(\frac{9 - y}{5}\right)\left(\frac{6}{11}\right) = \frac{9 - y}{5} - \frac{6}{11}$

Step 2: Eliminate the fractions by multiplying through by the least common denominator (LCD) of 55 (since 5 and 11 are the denominators):

$55 \times \left(1 + \left(\frac{9 - y}{5} \times \frac{6}{11}\right)\right) = 55 \times \left(\frac{9 - y}{5} - \frac{6}{11}\right)$

Simplifying the fractions: $55 + \left(55 \times \frac{9 - y}{5} \times \frac{6}{11}\right) = 11(9 - y) - 5(6)$

The term on the left simplifies as: [ 55 + 6(9 - y) =