Question

writing equations of parallel lines
what is the slope of the line that is parallel to the given line and passes through the given point?
what is the equation, in point - slope form, of the line that is parallel to the given line and passes through the given point?
what is the y - intercept of the line that is parallel to the given line and passes through the given point?

Explanation:

Step1: Find slope of given line

Let two points on the given line be $(-4,8)$ and $(4,6)$. Using the slope - formula $m=\frac{y_2 - y_1}{x_2 - x_1}$, we have $m=\frac{6 - 8}{4-(-4)}=\frac{-2}{8}=-\frac{1}{4}$. Parallel lines have equal slopes. So the slope of the parallel line is $-\frac{1}{4}$.

Step2: Write point - slope form

Let the given point be $(2,-4)$. The point - slope form of a line is $y - y_1=m(x - x_1)$. Substituting $m =-\frac{1}{4}$, $x_1 = 2$ and $y_1=-4$, we get $y+4=-\frac{1}{4}(x - 2)$.

Step3: Find y - intercept

Rewrite the point - slope form $y+4=-\frac{1}{4}(x - 2)$ in slope - intercept form $y=mx + b$.
\[

$$\begin{align*} y+4&=-\frac{1}{4}x+\frac{1}{2}\\ y&=-\frac{1}{4}x+\frac{1}{2}-4\\ y&=-\frac{1}{4}x-\frac{7}{2} \end{align*}$$

\]
The y - intercept $b =-\frac{7}{2}$.

Answer:

Slope: $-\frac{1}{4}$
Point - slope form: $y + 4=-\frac{1}{4}(x - 2)$
y - intercept: $-\frac{7}{2}$