# Equation of a Line Given Slope and a Point

### Linear Equations and Graphing

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

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
$$\color{blue}{y=-2,000x+30,000}$$