# Applications of Slope

Slope can be interpreted in many real-world relationships, such as speed, unit cost, and rates of change.
In real-world situations, a rate of change is often represented using the word per, as in kilometers per hour, dollars per hour, or liters per minute. The first quantity indicates the dependent variable, which is often represented by the variable $y$. The second quantity indicates the independent variable, which is often represented by the variable $x$, or by the variable $t$ if it represents time. The y-intercept, which is the value of $y$ where a graph touches or crosses the $y$-axis, represents the value of the independent variable at the start of the problem, where $x=0$ or $t=0$.
Step-By-Step Example
Interpreting Slope and $y$-Intercept from a Graph
Leila is walking to the library at a constant speed. The graph represents her distance, or $d$, in kilometers from the library after a number of hours, or $t$. Identify the slope and $y$-intercept.
Step 1
The graph goes through the point $(0, 6)$, so the $y$-intercept is 6.
Step 2

The rate of change is the slope of the line. To determine the slope, choose two points on the graph.

At time $t=0$, Leila is 6 kilometers from the library. This is shown on the graph as $(0, 6)$.

After 2 hours, at $t=2$, her distance from the library is zero kilometers. This is shown on the graph as $(2, 0)$.

Step 3
Use the two chosen points on the graph to calculate the slope.
\begin{aligned}m&=\frac{y_2-y_1}{x_2-x_1}\\&=\frac{0-6}{2-0}\\&=\frac{-6}{2}\\&=-3\end{aligned}
Solution

The $y$-intercept is 6. This means that when Leila starts walking and zero hours have passed, she is 6 kilometers from the library.

Leila's rate of change is –3 kilometers per hour. This means she is walking at a speed of 3 kilometers per hour. The rate is negative because her distance from the library is decreasing as she gets closer.

Step-By-Step Example
Interpreting Slope and $y$-Intercept from an Equation
A 20-liter bucket is being filled with water. The equation describes the volume of water in the bucket in liters, $y$, after $x$ minutes:
$y=2x+5$
Identify the slope and $y$-intercept of the line described by the equation, and describe their meanings. At what time will the bucket be full?
Step 1
The equation is in slope-intercept form, so the $y$-intercept is the value of $b$, or 5.
Step 2
The rate of change is the slope of the equation. The equation is in slope-intercept form, so the slope is the value of $m$, or 2.
Step 3
Determine when the bucket will be full. The bucket holds 20 liters. So, substitute 20 for $y$ in the equation and solve for $x$.
\begin{aligned}y&=2x+5\\20&=2x+5\\15&=2x\\7.5&=x\end{aligned}
Solution

The $y$-intercept is 5. This means that there were 5 liters of water already in the bucket at $x=0$ minutes.

The slope represents the number of liters added to the bucket each minute, which is 2 liters per minute.

When $y=20$, the value of $x$ is 7.5. So, the bucket will be full after 7.5 minutes.