To solve the problem, we need to find the solution to the system of equations by identifying the graph where the two lines intersect. The system of equations is:

1. $y = \frac{2}{3}x + 1$
2. $y = -\frac{2}{3}x - 1$

Steps to Solve:

1. Equation 1 ($y = \frac{2}{3}x + 1$):

   • Slope ($m$) is $\frac{2}{3}$ (line increases as $x$ increases).
   • $y$-intercept ($b$) is $1$.

2. Equation 2 ($y = -\frac{2}{3}x - 1$):

   • Slope ($m$) is $-\frac{2}{3}$ (line decreases as $x$ increases).
   • $y$-intercept ($b$) is $-1$.

3. Solution:

   • The solution to the system of equations is where the two lines intersect. Let’s calculate the intersection point algebraically: $\frac{2}{3}x + 1 = -\frac{2}{3}x - 1$ Combine like terms: $\frac{4}{3}x = -2$ Solve for $x$: $x = -\frac{3}{2}$ Substitute $x = -\frac{3}{2}$ into $y = \frac{2}{3}x + 1$: $y = \frac{2}{3}(-\frac{3}{2}) + 1 = -1 + 1 = 0$ Intersection point is $(-\frac{3}{2}, 0)$.

4. Verify on Graphs:

   • Look for the graph where the intersection point is approximately $(-1.5, 0)$.

Based on the provided graphs, the correct graph is the one that shows the two lines intersecting at approximately $(-1.5, 0)$. This appears to be the middle graph.

Let me know if you'd like a further explanation or have any questions! Here are five follow-up questions to deepen your understanding:

1. How can you verify a solution to a system of equations graphically and algebraically?
2. What role does the slope play in determining the angle of intersection between two lines?
3. What are some methods other than graphing to solve a system of linear equations?
4. How can you find the solution if the lines are parallel or coincident?
5. Why is it useful to interpret solutions graphically in real-world contexts?

Tip: When graphing linear equations, always identify the slope and $y$-intercept to quickly sketch the line.