The image shows a diagram of two intersecting lines forming four angles, each labeled with expressions involving $x$ and $y$:

• The top angle is labeled $(2y + 50)^\circ$,
• The right angle is labeled $(7x - 248)^\circ$,
• The bottom angle is labeled $(x + 44)^\circ$,
• The left angle is labeled $(5y - 17)^\circ$.

Since the lines intersect, the opposite angles are equal. Thus, we can set up the following two equations based on the fact that vertically opposite angles are equal:

1. $(2y + 50) = (x + 44)$ (top and bottom angles),
2. $(7x - 248) = (5y - 17)$ (right and left angles).

Let's solve these two equations step by step:

1. From $(2y + 50) = (x + 44)$, solve for $x$: $2y + 50 = x + 44$ $x = 2y + 50 - 44$ $x = 2y + 6$

2. Substitute $x = 2y + 6$ into the second equation $(7x - 248) = (5y - 17)$: $7(2y + 6) - 248 = 5y - 17$ Expand: $14y + 42 - 248 = 5y - 17$ Simplify: $14y - 206 = 5y - 17$ Subtract $5y$ from both sides: $9y - 206 = -17$ Add 206 to both sides: $9y = 189$ Solve for $y$: $y = \frac{189}{9} = 21$

3. Now that we have $y = 21$, substitute this value into $x = 2y + 6$ to find $x$: $x = 2(21) + 6 = 42 + 6 = 48$

Therefore, the solutions are:

$x = 48, \quad y = 21$

Would you like further details on this solution or another explanation?

Here are 5 related questions:

1. How do we know that opposite angles are equal when two lines intersect?
2. Can we use the same method to solve systems of equations in more complex situations?
3. How do we check if the values of $x$ and $y$ satisfy both equations?
4. What happens if one of the angles in a system is supplementary instead of equal?
5. How can we apply this concept to real-world problems involving angles?

Tip: When solving systems of equations, always start by simplifying one equation and then substitute into the other to avoid errors in algebraic manipulation.