The problem states that quadrilateral $ABCD$ is inscribed in a circle. The angles are given as follows: $\angle A = 5y$, $\angle B = 5x+11$, $\angle C = 7y$, and $\angle D = 8x$. We are asked to find the values of $x$ and $y$.

1. Problem Description

The problem states that quadrilateral $ABCD$ is inscribed in a circle. The angles are given as follows: $\angle A = 5y$, $\angle B = 5x+11$, $\angle C = 7y$, and $\angle D = 8x$. We are asked to find the values of $x$ and $y$.

2. Solution Steps

Since quadrilateral $ABCD$ is inscribed in a circle, opposite angles are supplementary, meaning that their sum is $180^\circ$. Therefore, we have the following two equations:

$\angle A + \angle C = 180^\circ$

$\angle B + \angle D = 180^\circ$

Substituting the given expressions for the angles, we get:

$5y + 7y = 180$

$5x + 11 + 8x = 180$

Simplify the equations:

$12y = 180$

$13x + 11 = 180$

Solve for $y$:

$y = \frac{180}{12}$

$y = 15$

Solve for $x$:

$13x = 180 - 11$

$13x = 169$

$x = \frac{169}{13}$

$x = 13$

3. Final Answer

$x = 13$

$y = 15$