The image presents a system of linear equations related to angles formed by intersecting lines. Here's the setup:

• The given angles are represented algebraically:
  • $6x + 7y^\circ$
  • $4x + 3y^\circ$
  • The angle of $90^\circ$ (right angle) is also provided.
  • The other angle is $40^\circ$.

Step-by-Step Approach:

1. Identify the relationships:

   • Vertically opposite angles are equal. Therefore, $6x + 7y = 90^\circ$ since it's opposite the 90-degree angle.
   • Adjacent angles along a straight line add up to 180°. So, $4x + 3y + 40^\circ = 180^\circ$.

2. Set up the system of equations:

   • Equation 1: $6x + 7y = 90$
   • Equation 2: $4x + 3y + 40 = 180$ simplifies to $4x + 3y = 140$.

Now we can solve the system of equations.

Let's solve this system for $x$ and $y$.The solution to the system of equations is:

• $x = 71$
• $y = -48$

If you need further explanation or details, feel free to ask!

Here are 5 additional questions that extend from this problem:

1. How can you verify that these values of $x$ and $y$ are correct?
2. What happens if one of the angles in the figure changes? How would the system of equations be affected?
3. Can you solve a similar system of equations using a different method, like substitution or elimination?
4. How do vertically opposite and supplementary angles help in solving geometry problems?
5. What other types of angle-related problems can be solved using systems of linear equations?

Tip: When solving systems of equations, always check if the equations are consistent and form a valid geometry before interpreting solutions like negative angles.