The image shows a graph with two intersecting lines, and the problem asks for the correct solution based on the graph. The graph contains an $x$-axis and a $y$-axis with labeled points, and the two lines intersect near the point $(2, 7)$.

Step-by-Step Analysis:

1. The graph shows two lines:
   • One with a positive slope crossing through $(0, 7)$ and $(2, 0)$.
   • The other with a negative slope crossing through $(0, -5)$ and intersecting at $(2, 0)$.

Interpretation:

The problem likely asks for the equations of the two lines or the point where they intersect.

Line 1:

This line appears to pass through $(0, 7)$ (the y-intercept) and $(2, 0)$. Using the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 7}{2 - 0} = \frac{-7}{2}$ The equation of the line using the slope-intercept form $y = mx + b$ is: $y = -\frac{7}{2}x + 7$

Line 2:

This line passes through $(0, -5)$ (the y-intercept) and intersects $(2, 0)$. Calculating the slope: $m = \frac{0 - (-5)}{2 - 0} = \frac{5}{2}$ The equation of this line is: $y = \frac{5}{2}x - 5$

Solution:

From the graph, it seems that the solution is the point of intersection of these two lines. To find the intersection point, solve the system of equations: $-\frac{7}{2}x + 7 = \frac{5}{2}x - 5$ Solve for $x$: $-\frac{7}{2}x - \frac{5}{2}x = -5 - 7$ $-\frac{12}{2}x = -12 \quad \Rightarrow \quad -6x = -12 \quad \Rightarrow \quad x = 2$

Now substitute $x = 2$ into one of the equations (e.g., $y = -\frac{7}{2}x + 7$): $y = -\frac{7}{2}(2) + 7 = -7 + 7 = 0$ Thus, the intersection point is $(2, 0)$.

The correct solution is the intersection point $(2, 0)$.

Would you like more details or clarifications on this?

Here are 5 related questions to expand on this topic:

1. What is the general method for finding the intersection of two lines algebraically?
2. How do you graph a linear equation using its slope and y-intercept?
3. Can two lines have more than one point of intersection? Why or why not?
4. What does the slope of a line tell you about its steepness and direction?
5. How do parallel and perpendicular lines behave in terms of their slopes?

Tip: When graphing lines, always find the y-intercept and slope to quickly draw the line and determine key points.