The problem asks for the equations of the two lines, $m$ and $n$, shown in the graph in slope-intercept form. Then it asks for the product of the slopes of the two lines and why the product is the value it is.

1. Problem Description

The problem asks for the equations of the two lines, $m$ and $n$, shown in the graph in slope-intercept form. Then it asks for the product of the slopes of the two lines and why the product is the value it is.

2. Solution Steps

First, we need to determine the slope and y-intercept for each line. The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

For line $m$:

We can choose two points on the line, for example, $(-3, 4)$ and $(3, 0)$.

The slope $m_m$ is given by the formula:

$m_m = \frac{y_2 - y_1}{x_2 - x_1}$

$m_m = \frac{0 - 4}{3 - (-3)} = \frac{-4}{6} = -\frac{2}{3}$.

The y-intercept of line $m$ is the point where the line crosses the y-axis, which appears to be at $(0, 2)$. So, $b_m = 2$.

Therefore, the equation of line $m$ is $y = -\frac{2}{3}x + 2$.

For line $n$:

We can choose two points on the line, for example, $(1, -1)$ and $(3, 3)$.

The slope $m_n$ is given by the formula:

$m_n = \frac{y_2 - y_1}{x_2 - x_1}$

$m_n = \frac{3 - (-1)}{3 - 1} = \frac{4}{2} = 2$.

The y-intercept of line $n$ is the point where the line crosses the y-axis, which appears to be at $(0, -3)$. So, $b_n = -3$.

Therefore, the equation of line $n$ is $y = 2x - 3$.

Next we compute the product of the slopes of lines $m$ and $n$.

Product of slopes = $m_m \cdot m_n = (-\frac{2}{3}) \cdot (2) = -\frac{4}{3}$.

The lines are not perpendicular to each other since the product of their slopes is not -

1.

3. Final Answer

The equation of line $m$ is $y = -\frac{2}{3}x + 2$.

The equation of line $n$ is $y = 2x - 3$.

The product of the slopes of lines $m$ and $n$ is $-\frac{4}{3}$.

The lines are not perpendicular because the product of their slopes is not -1.