Equation of a Line Given Slope and a Point

Sometimes we are asked to find the equation of a line given only the slope and a point on the line.

 
plane dividing

Your Turn:

An airplane begins descending at the rate of 2000 feet per minute. After 1 min it is 28,000 feet above the ground. Assume the plane continues at the same rate of descent. What is the equation of the line that represents the plane’s decent.

 
 

Solution:

We will start by using the slope-intercept form of line equation: $\color{blue}{y=mx+b}$
Where $x$ represents the time and $y$ represents the distance from the ground.

We are told that the plane is descending at the rate of 2000 feet per minute. That would represent the slope of the equation.

$\color{blue}{m=\frac{-2000 ft.}{1 min.}}$ $$
\begin{align*}
y&=\color{blue}mx+b\\
y&=\color{blue}{-2000}x+b
\end{align*}
$$

Now we need to find a coordinate that will be on the line of the equation. Since the airplane was 28,000 feet above the ground after 1 minute, our coordinates are $(1,28000)$.
$$
\begin{align*}
\color{blue}y&=-2000\color{blue}x+b\\
\color{blue}{28,000}&={-2,000}*\color{blue}1+b
\end{align*}
$$

Now we can solve for b.
$$
\begin{align*}
28,000&={-2,000}*1+b\\
28,000\color{blue}{+2000}&={-2,000}\color{blue}{+2000}+b\\
\color{blue}{30,000}&=b
\end{align*}
$$
Substitute into the equation $y=mx+b$ to determine the equation of the line formed
$$
\color{blue}{y=-2,000x+30,000}
$$