Linear Graphing & Slope

Given the points $(-3,5)$ and $(2,-5)$:

Let $x_1=-3$, $y_1=5$

Let $x_2=2$, $y_2=-5$

Substitute these values into the formula:

$m=(-5-5)/(2-(-3))$

$m=(-10)/(2+3)$

$m=-\frac{10}{5}$

$m=-2$

The slope of the line is $-2$.

In this form:

- $m$ represents the slope, which is the rate of change. Here, $7.50$ is the cost per T-shirt.

- $b$ represents the y-intercept, which is the value of $C$ when $n=0$. When $n=0$ (no T-shirts are printed), $C=30$. This initial cost of $$30$ is the fixed setup fee that must be paid regardless of how many T-shirts are ordered.

Therefore, $30$ represents the fixed setup fee.

We are given that the slope $m=4$. So the equation starts as $y=4x+b$.

We are also given that the line passes through the point (2, 3). This means that when $x=2$, $y=3$. We can substitute these values into the equation to find $b$:

$3=4(2)+b$

$3=8+b$

Now, solve for $b$:

$3-8=b$

$-5=b$

So, the y-intercept is $-5$.

Substitute $m=4$ and $b=-5$ back into the slope-intercept form:

$y=4x-5$

Line P passes through the points (1, 5) and (3, 11). Use the slope formula $m=(y_2-y_1)/(x_2-x_1)$:

$m_P=(11-5)/(3-1)$

$m_P=\frac{6}{2}$

$m_P=3$

Next, find the slope of Line Q.

Line Q is represented by the equation $y=3x-2$. This equation is already in slope-intercept form, $y=mx+b$, where $m$ is the slope.

From the equation, the slope of Line Q, $m_Q$, is $3$.

Now, compare the slopes:

Slope of Line P ($m_P$) is $3$.

Slope of Line Q ($m_Q$) is $3$.

Since $3=3$, the slopes of Line P and Line Q are equal.