Can you determine the slope of the line with an \begin{align*}x\end{align*}-intercept of 4 and \begin{align*}y\end{align*}-intercept of –3?

### Slope of Lines from Two Points

The **slope** of a line is the steepness, slant or gradient of that line. Slope is defined as \begin{align*}\frac{\text{rise}}{\text{run}}\end{align*} (rise over run) or \begin{align*}\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}\end{align*} (change in \begin{align*}y\end{align*} over change in \begin{align*}x\end{align*}). Whatever definition of slope is used, they all mean the same. The slope of a line is represented by the letter ‘\begin{align*}m\end{align*}’ and its value is a real number.

You can calculate the slope of a line by using the coordinates of two points on the line. Consider a line that passes through the points \begin{align*}A (-6, -4)\end{align*} and \begin{align*}B (3, -8)\end{align*}. The slope of this line can be determined by finding the change in \begin{align*}y\end{align*} over the change in \begin{align*}x\end{align*}.

The formula that is used is \begin{align*}m=\frac{y_2-y_1}{x_2-x_1}\end{align*} where ‘\begin{align*}m\end{align*}’ is the slope, \begin{align*}(x_1,y_1)\end{align*} are the coordinates of the first point and \begin{align*}(x_2,y_2)\end{align*} are the coordinates of the second point. The choice of the first and second point will not affect the result.

\begin{align*}& A \begin{pmatrix} x_1, & y_1 \\ -6, & -4 \end{pmatrix} \quad B \begin{pmatrix} x_2, & y_2 \\ 3, & -8 \end{pmatrix}&& \text{Label the points to indicate the first and second points.}\\ & m=\frac{y_2-y_1}{x_2-x_1} && \text{Substitute the values into the formula.}\\ & m=\frac{-8--4}{3--6} && \text{Simplify the values (if possible)}\\ & m=\frac{-8+4}{3+6} && \text{Evaluate the numerator and the denominator}\\ & m=\frac{-4}{9} && \text{Reduce the fraction (if possible)}\end{align*}

#### Let's practice finding the slope given two points in the following problems:

- (–3, –8) and (5, 8).

To determine the slope of a line from two given points, the formula \begin{align*}m=\frac{y_2-y_1}{x_2-x_1}\end{align*} can be used. Don’t forget to designate your choice for the first and the second point. Designating the points will reduce the risk of entering the values in the wrong location of the formula.

\begin{align*}& \begin{pmatrix} x_1, & y_1 \\ -3, & -8 \end{pmatrix} \quad \begin{pmatrix} x_2, & y_2 \\ 5, & 8 \end{pmatrix}\\ & \text{Substitute the values into the formula} && m=\frac{y_2-y_1}{x_2-x_1}\\ & && m=\frac{8--8}{5--3}\\ & \text{Simplify} && m=\frac{8+8}{5+3}\\ & \text{Calculate} && m=\frac{16}{8}\\ & \text{Simplify} && m=2\end{align*}

- \begin{align*}(9, 5)\end{align*} and \begin{align*}(-1, 6)\end{align*}.

\begin{align*}& \begin{pmatrix} x_1, & y_1 \\ 9, & 5 \end{pmatrix} \quad \begin{pmatrix} x_2, & y_2 \\ -1, & 6 \end{pmatrix}\\ & m =\frac{y_2-y_1}{x_2-x_1}\\ & m =\frac{6-5}{-1-9}\\ & m =-\frac{1}{10}\end{align*}

- \begin{align*}(-2, 7)\end{align*} and \begin{align*}(-3, -1)\end{align*}.

\begin{align*}& \begin{pmatrix} x_1, & y_1 \\ -2, & 7 \end{pmatrix} \quad \begin{pmatrix} x_2, & y_2 \\ -3, & -1 \end{pmatrix}\\ & m =\frac{y_2-y_1}{x_2-x_1}\\ & m =\frac{-1-7}{-3--2}\\ & m =\frac{-1-7}{-3+2}\\ & m =\frac{-8}{-1}\\ & m =8\end{align*}

### Examples

#### Example 1

Earlier, you were asked to find the slope of a line with an \begin{align*}x\end{align*}-intercept of 4 and \begin{align*}y\end{align*}-intercept of –3.

\begin{align*}& \begin{pmatrix} x_1, & y_1 \\ 4, & 0 \end{pmatrix} \quad \begin{pmatrix} x_2, & y_2 \\ 0, & -3 \end{pmatrix} && \text{Express the} \ x- \text{and} \ y \text{-intercepts as coordinates of a point.}\\ & m =\frac{y_2-y_1}{x_2-x_1}\\ & m =\frac{-3-0}{0-4}\\ & m =\frac{-3}{-4}\\ & m =\frac{3}{4}\end{align*}

#### Example 2

Calculate the slope of the line that passes through the points:

(5, –7) and (16, 3)

- \begin{align*}& \begin{pmatrix} x_1, & y_1 \\ 5, & -7 \end{pmatrix} \quad \begin{pmatrix} x_2, & y_2 \\ 16, & 3 \end{pmatrix} && \text{Designate the points as to the first point and the second point.}\\ & m =\frac{y_2-y_1}{x_2-x_1}\\ & m =\frac{3--7}{16-5} && \text{Fill in the values}\\ & m =\frac{3+7}{16-5} && \text{Simplify the numerator and denominator (if possible)}\\ & m =\frac{10}{11} && \text{Calculate the value of the numerator and the denominator}\end{align*}

The slope is \begin{align*}\frac{10}{11}\end{align*}.

#### Example 3

Calculate the slope of the line that passes through the points:

(–6, –7) and (–1, –4)

- \begin{align*}& \begin{pmatrix} x_1, & y_1 \\ -6, & -7 \end{pmatrix} \quad \begin{pmatrix} x_2, & y_2 \\ -1, & -4 \end{pmatrix} && \text{Designate the points as to the first point and the second point.}\\ & m =\frac{y_2-y_1}{x_2-x_1}\\ & m =\frac{-4--7}{-1--6} && \text{Fill in the values}\\ & m =\frac{-4+7}{-1+6} && \text{Simplify the numerator and denominator (if possible)}\\ & m =\frac{3}{5} && \text{Calculate the value of the numerator and the denominator}\end{align*}

The slope is \begin{align*}\frac{3}{5}\end{align*}.

#### Example 4

Calculate the slope of the line that passes through the points:

(5, –12) and (0, –6)

- \begin{align*}& \begin{pmatrix} x_1, & y_1 \\ 5, & -12 \end{pmatrix} \quad \begin{pmatrix} x_2, & y_2 \\ 0, & -6 \end{pmatrix} && \text{Designate the points as to the first point and the second point.}\\ & m =\frac{y_2-y_1}{x_2-x_1}\\ & m =\frac{-6--12}{0-5} && \text{Fill in the values}\\ & m =\frac{-6+12}{0-5} && \text{Simplify the numerator and denominator (if possible)}\\ & m =\frac{6}{-5} && \text{Calculate the value of the numerator and the denominator}\\ & m =-\frac{6}{5} \end{align*}

The slope is \begin{align*}-\frac{6}{5}\end{align*}.

#### Example 5

The local Wine and Dine Restaurant has a private room that can serve as a banquet facility for up to 200 guests. When the manager quotes a price for a banquet she includes the cost of the room rent in the price of the meal. The price of a banquet for 80 people is $900 while one for 120 people is $1300.

- i) Plot a graph of cost versus the number of people.
- ii) What is the slope of the line and what meaning does it have for this situation?

- The domain for this situation is \begin{align*}N\end{align*}. However, to demonstrate the slope and its meaning, it is more convenient to draw the graph as \begin{align*}x \ \varepsilon \ R\end{align*} instead of showing just the points on the Cartesian grid. The \begin{align*}x\end{align*}-axis has a scale of 10 and the \begin{align*}y\end{align*}-axis has a scale of 100. The slope can be calculated by counting to determine \begin{align*}\frac{\text{rise}}{\text{run}}\end{align*}.
- From the point to the left, run four spaces (40) in a positive direction and move upward four spaces (400) in a positive direction.
- \begin{align*}m&=\frac{\text{rise}}{\text{run}}\\ m&=\frac{400}{{\color{blue}40}}\\ m &= \frac{10}{{\color{blue}1}}\\ m &= \frac{10 \ dollars}{{\color{blue}1 \ person}}\end{align*}
- The slope represents the cost of the meal for each person. It will cost $10 per person for the meal.

### Review

Calculate the slope of the line that passes through the following pairs of points:

- (3, 1) and (–3, 5)
- (–5, –57) and (5, –5)
- (–3, 2) and (7, –1)
- (–4, 2) and (4, 4)
- (–1, 5) and (4, 3)
- (0, 2) and (4, 1)
- (12, 15) and (17, 3)
- (2, –43) and (2, –14)
- (–16, 21) and (7, 2)

The cost of operating a car for one month depends upon the number of miles you drive. According to a recent survey completed by drivers of midsize cars, it costs $124/month if you drive 320 miles/month and $164/month if you drive 600 miles/month.

- Plot a graph of distance/month versus cost/month.
- What is the slope of the line and what does it represent?

A Glace Bay developer has produced a new handheld computer called the Blueberry. He sold 10 computers in one location for $1950 and 15 in another for $2850. The number of computers and the cost forms a linear relationship.

- Plot a graph of number of computers sold versus cost.
- What is the slope of the line and what does it represent?

Shop Rite sells one-quart cartons of milk for $1.65 and two-quart cartons for $2.95. Assume there is a linear relationship between the volume of milk and the price.

- Plot a graph of volume of milk sold versus cost.
- What is the slope of the line and what does it represent?

Some college students, who plan on becoming math teachers, decide to set up a tutoring service for high school math students. One student was charged $25 for 3 hours of tutoring. Another student was charged $55 for 7 hours of tutoring. The relationship between the cost and time is linear.

- Plot a graph of time spent tutoring versus cost.
- What is the slope of the line and what does it represent?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.2.