1.3 The Slope of a Line

Think of what you have learned about linear functions in previous courses. How would you describe the slope of a line? It is the $m$ in the slope-intercept formula $y=mx+b$. You may have memorized that the slope of a line is “rise over run.” The slope tells us the linear relationship between two variables. It is the rate of change per unit change in $x$. If the slope is positive, the input variable and output variable have a positive relationship. If the slope is negative, the input variable and output variable have a negative relationship.

Mathematically, the slope of line $L$ is not a vertical line and has two distinct points $P_1$ and $P_2$, such that $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$, where $x_1\ne x_2$, is

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{y_1-y_2}{x_1-x_2}=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$

As long as the points used to calculate the slope are two distinct points on the same line, then it does not matter which two points you choose. When there is no change in the values of $x$ for two distinct points, but there is a change in the values of $y$ for those points, then the line is vertical, and the slope is, by convention, undefined. If there is a change in the values of $x$ for two points but not in the values of $y$, then the line is horizontal, and the slope is 0.

Suppose that you purchased four gallons of gasoline for $8, and your friend purchased six gallons of gasoline for $12 at the same gas station at the same time. Assume that the cost of purchasing gasoline is a linear function of the gallons of gasoline purchased. What is the price per gallon of gasoline (the slope of the function)?

Your friend purchased two gallons more than you did and she paid $4 more for those extra two gallons. This means that:

$m=\text{Price per gallon}=\frac{\mathrm{\$}4}{2}=\mathrm{\$}2$

Examples

Let’s practice finding the slope of a line given two points.

Example 1

Suppose you have two points, (2, 3) and (5, 9). What is the slope of the line passing through these two points?

$m=\frac{9-3}{5-2}=\frac{6}{3}=2$

Example 2

Erin and Todd are part-time employees of your company. Erin worked 12 hours this week and earned $150, while Todd worked 8 hours this week and earned $100. If both of them earn the same hourly wage, what is their hourly wage?

Notice that the hourly wage is the rate of change in wage per hour of work. Therefore, we can use the slope equation to solve this problem. The $x$ variable represents hours worked, and the $y$ variable represents total wages earned, so:

$m=\text{Hourly Wage}=\frac{150-100}{12-8} =\frac{50}{4}=12.5$

Example 3

Two of your employees recently requested your company to print their business cards. Amy ordered 200 business cards at a total cost of $20, and Adam ordered 500 business cards at a total cost of $35. Given that the cost is a linear function of the number of cards printed, what is the printing cost per business card?

$m=\text{Printing Cost per Card}=\frac{35-20}{500-200}=\frac{15}{300}=0.05$