Solution:

(a) To solve the system of equations $y = x + 2$ and $y = 2x$ using the graph:

The solution is the intersection point of the two lines. Observing the graph:

• The purple line represents $y = 2x$.
• The green line represents $y = x + 2$.

These lines intersect at the point $(2, 4)$.
Thus, the solution to the simultaneous equations is: $x = 2, \quad y = 4.$

(b) To analyze further:

1. Substituting $x = 2$:

   • In $y = x + 2$:
     $y = 2 + 2 = 4$

   • In $y = 2x$:
     $y = 2(2) = 4$

   What do you notice?

   • The value of $y$ is 4 in both equations, confirming consistency.

2. Comparison with part (a):

   • The value of $y$ found through substitution matches the value at the intersection point determined graphically. Both give $y = 4$.

Let me know if you'd like a detailed explanation for any step or further questions. Here are some related explorations:

1. How do we solve simultaneous equations algebraically?
2. What happens if the two equations are parallel?
3. How do we graph non-linear simultaneous equations (e.g., one line and one parabola)?
4. How can systems of equations be solved using matrices?
5. How does the solution change if we modify the slopes or intercepts of the lines?

Tip: Always cross-check graphical solutions with algebraic substitution for accuracy!