The image shows two linear equations graphed on a coordinate plane. The equations are labeled as $y = x + 7$ and $y = 6x + 2$, which intersect at a point labeled $P$.

Below the graph, there are multiple-choice options where each option lists two equations. Your task is to identify the set of equations that match the two lines in the graph.

• Line 1: The line labeled $y = x + 7$ has a slope of $1$ and a y-intercept of $7$.
• Line 2: The line labeled $y = 6x + 2$ has a slope of $6$ and a y-intercept of $2$.

Let's compare these equations to the given options:

1. Option 1: $y = 5x + 3$ and $y = 9x - 1$
   These do not match the given lines in slope or intercept.

2. Option 2: $y = 4x + 2$ and $y = 6x + 2$
   This is close, but the first equation doesn’t match $y = x + 7$.

3. Option 3: $y = -2x + 10$ and $y = 8x + 1$
   These equations have completely different slopes and intercepts.

4. Option 4: $y = 7x - 5$ and $y = -5x + 15$
   These also do not match either line in the graph.

Since none of the options fully match the equations given on the graph ($y = x + 7$ and $y = 6x + 2$), it's possible that there is a mismatch in the multiple-choice options. Would you like to explore further or provide any clarifications?

Follow-up questions:

1. How do you determine the slope and intercept from a linear equation?
2. How do two lines' slopes help in determining if they are parallel, perpendicular, or intersecting?
3. Can you solve for the point of intersection between two lines algebraically?
4. Why is it important to identify the y-intercept when graphing a linear equation?
5. How can you derive an equation of a line given its graph?

Tip: The slope is the change in $y$ over the change in $x$, which determines how steep a line is.