This is a geometry problem involving parallel lines and alternate interior angles. Let’s break it down step by step to find $x$.

We have three parallel lines, and the transversal cuts through them. Based on the properties of angles formed by parallel lines, the angles on opposite sides of the transversal are either corresponding or alternate interior angles.

Step 1: Set up the equations

1. The angle $(5x - y)^\circ$ is an alternate interior angle with the 150° angle (since they are on opposite sides of the transversal). Therefore, they must be equal: $5x - y = 150$

2. Similarly, the angle $(5x + y)^\circ$ is an alternate interior angle with the 130° angle: $5x + y = 130$

Step 2: Solve the system of equations

We now have two equations:

1. $5x - y = 150$
2. $5x + y = 130$

Add these two equations together: $(5x - y) + (5x + y) = 150 + 130$ This simplifies to: $10x = 280$ Solving for $x$: $x = 28$

So, $x = 28$.

Would you like further clarification on any part of the solution, or do you have any other questions?

Relative Questions:

1. How do alternate interior angles behave with parallel lines?
2. How can you set up equations for unknowns in geometry problems involving angles?
3. What are corresponding angles in parallel lines, and how do they relate to alternate interior angles?
4. How can you use systems of equations to solve for two unknown variables?
5. Why do the sum of angles on a straight line always equal 180°?

Tip:

In geometry problems involving parallel lines and transversals, always look for relationships like alternate interior angles or corresponding angles to set up equations.