More on Slope and Equations

Elizabeth Pople

20 More on Slope and Equations

Topics Covered:^[1]

In case you missed something in class, or just want to review a specific topic covered in this Module, here is a list of topics covered:

Slope and Equations
Graphing and Slope
Graphing with slope and intercepts
Key Concepts

Slope and Equations

Now that we’ve defined slope and looked at the Standard form of a line (Ax + By = C), there are two other forms of a linear equation in addition to the Standard form:

Point Slope Form

Slope Intercept Form

y − y₁= m(x − x₁)

y = mx + b

where m=slope (x₁,y₁) is a point on the line.

where m = slope and b = y-intercept.

What if all you are given is 2 points of a line? We know from the previous section that you could use them to determine slope, but this process will also allow us to find out what the linear equation would look like.

The Slope of a Line, m, represents the vertical change in y over the horizontal change in x. Given two points, (x₁,y₁) and (x₂,y₂), the following formula determines the slope of a line containing these points:

m = $\displaystyle\frac{y_2 - y_1}{x_2 - x_1}$

Slope also indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.

Suppose we have a line that has slope m and that contains some specific point (x₁,y₁) and some other point, which we will just call (x,y). We can write the slope of this line and then change it to a different form.

Steps

Algebraic

Equation

m = $\displaystyle\frac{y - y_1}{x - x_1}$

Multiply both sides of the equation by x − x₁.

m(x – x₁) = $\displaystyle (\frac{y - y_1}{x - x_1})(x - x_1)$

Simplify.

m(x – x₁) = y – y₁

Rewrite the equation with the y terms on the left.

y – y₁= m(x – x₁)

Point Slope Form

Given one point and the slope, the point-slope formula will lead to the equation of a line:

y − y₁= m (x − x₁)

What happens if you only have two points? Can we find the slope with just two points? Yes, using the steps we looked at in the previous section. Once we have the slope, we can use it and one of the given points to find the equation. The following example gives two points, so you need to solve for the slope first and then use one of the points.

How to find the equation with 2 Points:

Find the slope using the given points. m = $\displaystyle\frac{y_2 - y_1}{x_2 - x_1}$
Choose one point.
Substitute the values into the point-slope form: y − y₁= m (x − x₁)
Write the equation in slope-intercept form.

Try it!

Find the equation of the line passing through the points (3,4) and (0,−3). Write the final equation in slope-intercept form.

Solution (click to reveal)

First, we calculate the slope using the slope formula and two points.

$\displaystyle m =\frac{-3 - 4}{0 - 3}$

= $\displaystyle\frac{-7}{-3}$

= $\displaystyle\frac{7}{3}$

Next, we use the point-slope formula with the slope of $\displaystyle\frac{7}{3}$ , and either point. Let’s pick the point (3,4) for (x₁,y₁).

y – 4 = $\displaystyle\frac{7}{3}(x-3)$

Distribute the $\displaystyle\frac{7}{3}$

y – 4 = $\displaystyle\frac{7}{3}x$ – 7

y = $\displaystyle\frac{7}{3}x$ – 3

The equation is written as y = $\displaystyle\frac{7}{3}x$ – 3. (This is known as the Slope-intercept form.)

Analysis

To prove that either point can be used, let us use the second point (0,−3) and see if we get the same equation.

y – (-3) = $\displaystyle\frac{7}{3}(x - 0)$

y + 3 = $\displaystyle\frac{7}{3}x$

y = $\displaystyle\frac{7}{3}x$ – 3

We see that the same line will be obtained using either point. This makes sense because we used both points to calculate the slope.

Find an equation of a line that contains the points (−3,5) and (−3,4). Write the equation in slope-intercept form.

Solution (click to reveal)

Again, the first step will be to find the slope.

m = $\displaystyle\frac{y_2 - y_1}{x_2 - x_1}$

m = $\displaystyle\frac{4 - 5}{-3 - (-3)}$

m = $\displaystyle\frac{-1}{0}$

The Slope is undefined.

This tells us it is a vertical line. Both of our points have an x-coordinate of −2. So our equation of the line is x = −2. Since there is no y, we cannot write it in slope-intercept form.

You may want to sketch a graph using the two given points. Does your graph agree with our conclusion that this is a vertical line?

Try it! – How to Find the Equation of a Line Given Two Points

Find an equation of a line that contains the points (−3,−1) and (2,−2). Write the equation in slope-intercept form.

Solution (click to reveal)

Steps

Algebraic

Step 1. Find the slope using the given points

Find the slope of the line through (-3, -1) and (2, -2).

$\displaystyle \boldsymbol{m = \frac{y_2 - y_1}{x_2 - x_1}}$
$\displaystyle\boldsymbol{ m = \frac{-2 - (-1)}{2 - (-3)}}$
$\displaystyle \boldsymbol{m = - \frac{1}{5}}$

$\displaystyle \boldsymbol{\textcolor{mypurple1}{m = - \frac{1}{5}}}$

Step 2. Choose one point. Choose either point.

$\displaystyle \boldsymbol{\textcolor{myred1}{(x_1, y_1) = (2, -2)}}$

Step 3. Substitute the values into the point-slope form, $\displaystyle y - y_1 = m(x - x_1)$ .

Simplify.

$\displaystyle \boldsymbol{y - {\color{myred1}{y_1}} = {\color{mypurple1}{ m}}(x - {\color{myred1}{x_1}})}$

$\displaystyle\boldsymbol{ y - {\color{myred1}{(-2) }}= \textcolor{mypurple1}{- \frac{1}{5}}(x - {\color{myred1}{ 2)}}}$

$\displaystyle\boldsymbol{ y + 2 = \frac{-1}{5}x + \frac{2}{5}}$

Step 4. Write the equation in slope-intercept form.

$\displaystyle \boldsymbol{y = -\frac{1}{5}x - \frac{8}{5}}$

In the examples above, you will note that the form of the equation we get after using point-slope form is: y = mx+b. where m=slope and b = y-intercept. Practice identifying them below.

Try it! – Identifying the Slope and y-intercept of a Line Given an Equation

Identify the slope and y-intercept, given the equation y = – $\displaystyle\frac{3}{4}x$ – 4

Solution (click to reveal)

As the line is in y = mx+b form, the given line has a slope of m = $\displaystyle -\frac{3}{4}$ . The y-intercept is b = −4.

Analysis

The y-intercept is the point at which the line crosses the y-axis. On the y-axis, x=0. We can always identify the y-intercept when the line is in slope-intercept form, as it will always equal b. Or, just substitute x=0 and solve for y.

We can easily determine the slope and intercept of a line if the equation is written in slope-intercept form, y = mx+b. Now we will do the reverse—we will start with the slope and y-intercept and use them to find the equation of the line.

How To: Find the equation given a slope and a point

Identify the slope.
Identify the point.
Substitute the values into the point-slope form, y − y₁ = m(x − x₁).
Write the equation in slope-intercept form.

Try it! – Finding the Equation of a Line Given the Slope and One Point

Write the equation of the line with slope m = −3 and passing through the point (4,8). Then solve the equation for y.

Solution (click to reveal)

Using the point-slope formula, substitute −3 for m and the point (4,8) for (x₁,y₁).

y − y₁= m(x − x₁)

y − 8 = −3(x − 4)

y − 8 = −3x + 12

y = −3x + 20

Analysis

Note that any point on the line can be used to find the equation. If done correctly, the same final equation will be obtained.

Find the equation of a line with slope m = – $\displaystyle\frac{2}{5}x$ and containing the point (10, −5).

Solution (click to reveal)

y = – $\displaystyle\frac{2}{5}x$ – 1

Find the equation of the line shown in the graph.

This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 3, negative 6), (0, negative 4), (3, negative 2), and (6, 0).

Solution (click to reveal)

We need to find the slope and y-intercept of the line from the graph so we can substitute the needed values into the slope-intercept form, y=mx+b

To find the slope, we choose two points on the graph.

The y-intercept is (0,−4) and the graph passes through (3,−2).

Steps

Algebraic

Find the slope by counting the rise and run.

$m = \displaystyle\frac{\text{rise}}{\text{run}}$

Use the graph to identify the points and count.

$m = \displaystyle\frac{2}{3}$

Find the y-intercept.

$\textit{y-intercept } (0,\,-4)$

Substitute the values into y=mx+b

$\boldsymbol{ y = \color{myred1}{m}x + \color{mypurple1}{b}}$

Solution

$y = \displaystyle\frac{2}{3}x - 4$

Find an equation of a horizontal line that contains the point (−2,−6). Write the equation in slope-intercept form.

Solution (click to reveal)

Every horizontal line has slope 0. We can substitute the slope and points into the point-slope form, y − y₁ = m(x − x₁).

Steps

Algebraic

Identify the slope.

$\boldsymbol{m = \color{mypurple1}{0}}}$

Identify the point.

$\displaystyle\boldsymbol{\binom{{\color{myred1}{x_1}},\; {\color{myred1}{y_1} }} {{\color{myred1}{-2}},\; {\color{myred1}{-6} }}}$

Substitute the values into

y − y₁= m(x − x₁)

$\displaystyle\boldsymbol{y - {\color{myred1}{y_1}} = {\color{mypurple1}{m}}(x - {\color{myred1}{x_1}})}$
$\displaystyle\boldsymbol{y - {\color{myred1}{(-6)}} = {\color{mypurple1}{0}}(x - {\color{myred1}{(-2)}})}$

Simplify.

$\boldsymbol{y + 6 = 0}$

$\boldsymbol{y = -6}$

Write in slope-intercept form.

It is in y-form, but could be written y = 0x − 6.

Did we end up with the form of a horizontal line, y = a?

Graphing and Slope

We can also graph a line when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line.

Try it! – How to graph a Line Given a Point and the Slope

Graph the line passing through the point (1,−1) whose slope is m = $\displaystyle\frac{3}{4}$ .

Solution (click to reveal)

Steps

Algebraic

Plot the given point (1, -1).

Step 1 is to plot the given point. Plot (1, negative 1). The figure then shows the graph of a point on the x y-coordinate plane. The x-axis runs from negative 2 to 6. The y-axis runs from negative 2 to 4. The point (1, negative 1) is plotted and labeled with its coordinates.

Use the slope formula $\displaystyle m = \frac{\text{rise}}{\text{run}}$ to identify the rise and the run.

$\begin{align*} \displaystyle m &= \frac{3}{4} \\ \frac{rise}{run} &= \frac{3}{4} \\ rise &= 3 \\ run &= 4 \end{align*}$

Starting at the given point (1, -1), count out the rise ( up 3 units) and run (right 4 units) to mark the second point.

Step 3 is to start at the given point and count out the rise and run to mark the second point. Start at (1, negative 1) and count the rise and the run. Up 3 units, right 4 units. The figure then shows the graph of three points connected by two straight line segments on the x y-coordinate plane. The x-axis runs from negative 2 to 6. The y-axis runs from negative 2 to 4. The points (1, negative 1), (1, 2), and (5, 2) are plotted. A vertical line segment connects (1, negative 1) to (1, 2) and is labeled 3. A horizontal line segment connects (1, 2) to (5, 2) and is labeled 4.

Connect the two points with a line.

You can check your work by finding a third point. Since the slope is m = $\displaystyle\frac{3}{4}$ it can also be written as m = $\displaystyle\frac{-3}{-4}$ (negative divided by negative is positive!). Go back to (1,−1) and count out the rise, −3, and the run, −4.

Graph the line passing through the point (2,−2) with the slope m = $\displaystyle\frac{4}{3}$

Solution (click to reveal)

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (2, negative 2) and (5, 2).

Graph the line passing through the point (−2,3) with the slope m = $\displaystyle\frac{1}{4}$

Solution (click to reveal)

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (negative 2, 3) and (2, 4).

Graph a line given a point and the slope.

Plot the given point.
Use the slope formula m = $\displaystyle\frac{\text{rise}}{\text{run}}$ to identify the rise and the run.
Starting at the given point, count out the rise and run to mark the second point.
Connect the points with a line.

Graphing with Slope and Intercept

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using one point and the slope of the line. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines.

Let’s look at the graph of the equation y = $\displaystyle\frac{1}{2}x$ + 3 and find its slope and y-intercept.

The red lines in the graph show us that the rise is 1 and the run is 2. Substituting into the slope formula:

m = $\displaystyle\boldsymbol{\frac{\text{rise}}{\text{run}}}$

m = $\displaystyle \boldsymbol{\frac{1}{2}}$

The y-intercept is (0,3). Look at the equation of this line:

$\displaystyle\boldsymbol{y = {\color{myred1}{\frac{1}{2}}}x + {\color{mypurple1}{3}}}$

Look at the slope and y-intercept.

$\displaystyle\boldsymbol{\text{slope } m = {\color{myred1}{\frac{1}{2}}} \text{ and y-intercept } (0,\, {\color{mypurple1}{3}})}$

When a linear equation is solved for y, the coefficient of the x term is the slope, and the constant term is the y-coordinate of the y-intercept. We say that the equation y = $\displaystyle\frac{1}{2}x$ + 3 is in slope–intercept form. Sometimes the slope–intercept form is called the “y-form.”

$\displaystyle\boldsymbol{m = {\color{myred1}{\frac{1}{2}}};\ \text{y-intercept is } (0,\, {\color{mypurple1}{3}})}$

$\displaystyle\boldsymbol{y = {\color{myred1}{\frac{1}{2}}}x + {\color{mypurple1}{3}}}$

$\displaystyle\boldsymbol{y = {\color{myred1}{m}}x + {\color{mypurple1}{b}}}$

Slope-Intercept Form of an Equation of a Line

The slope–intercept form of an equation of a line with slope m and y-intercept (0, b) is y= mx + b.

Let’s practice finding the values of the slope and y-intercept from the equation of a line.

Try It!

Identify the slope and y-intercept of the line from the equation:

a. y = − $\displaystyle\frac{4}{7}x$ − 2

b. x + 3y = 9

Solution A (click to reveal)

a. We compare our equation to the slope–intercept form of the equation.

Steps

Algebraic

Write the slope–intercept form of the equation of the line.

$\displaystyle\boldsymbol{y = {\color{myred1}{m}}x + {\color{mypurple1}{b}}}$

Write the equation of the line.

$\displaystyle\boldsymbol{y = {\color{myred1}{-\frac{4}{7}}}x {\color{mypurple1}{- 2}}}$

Identify the slope.

$\displaystyle\boldsymbol{m = {\color{myred1}{-\frac{4}{7}}}}$

Identify the y-intercept.

$\displaystyle\boldsymbol{\text{y-intercept is } (0,\, {\color{mypurple1}{-2}})}$

Solution B (click to reveal)

b. When an equation of a line is not given in slope–intercept form, our first step will be to solve the equation for y.

Steps

Algebraic

Solve for y.

$\displaystyle\boldsymbol{x + 3y = 9}$

Subtract x from each side.

$\displaystyle\boldsymbol{3y = -x + 9}$

Divide both sides by 3.

$\displaystyle\boldsymbol{ \frac{3y}{3} = \frac{-x + 9}{3}}$

Simplify.

$\displaystyle\boldsymbol{y = -\frac{1}{3}x + 3}$

Write the slope–intercept form of the equation of the line.

$\displaystyle\boldsymbol{y = {\color{myred1}{m}}x + {\color{mypurple1}{b}}}$

Write the equation of the line.

$\displaystyle\boldsymbol{y = {\color{myred1}{-\frac{1}{3}}}x + {\color{mypurple1}{3}}}$

Identify the slope.

$\displaystyle\boldsymbol{m = {\color{myred1}{-\frac{1}{3}}}}$

Identify the y-intercept.

$\displaystyle\boldsymbol{\text{y-intercept is } (0,\, {\color{mypurple1}{3}})}$

Identify the slope and y-intercept from the equation of the line.

a. y = − $\displaystyle\frac{4}{3}x$ + 1

b. 3x+2y=12

Solution (click to reveal)

a. m = − $\displaystyle\frac{4}{3}$ ; (0, 1)

b. m = − $\displaystyle\frac{3}{2}$ ; (0, 6)

We have graphed a line using the slope and a point. Now that we know how to find the slope and y-intercept of a line from its equation, we can use the y-intercept as the point, and then count out the slope from there.

Graph the line of the equation y = −x + 4 using its slope and y-intercept.

Solution (click to reveal)

Steps

Algebraic

Equation

y = mx + b

The equation is in slope–intercept form.

y = −x + 4

Identify the slope and y-intercept.

m = −1

y-intercept is (0, 4)

Plot the y-intercept.

See the graph.

Identify the rise over the run.

m = $\displaystyle\frac{-1}{1}$

Count out the rise and run to mark the second point. Draw the line as shown in the graph.

rise −1, run 1
Graph showing a line running from quadrant 2 down through 1 on a left to right slant. Points (0,4) (1,3) are marked.

Graph the line of the equation y = −x − 3 using its slope and y-intercept.

Solution (click to reveal)

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 3) and (1, negative 4).

Now that we have graphed lines by using the slope and y-intercept, let’s summarize all the methods we have used to graph lines.

Point-Plotting

Slope-Intercept

Intercepts

Recognize Vertical and Horizontal Lines

Create a blank table with two columns and four rows. The first row is labeled “x” and “y”. Find three points. Plot the points, make sure they line up, them draw the line.

y = mx + b

Find the slope and y-intercept, Start at the y-intercept, then count the slope to get a second point.

Create a blank table with two columns and four rows. The first row is labeled “x” and “y”. Plug a zero in the first row in the x column. In the second row there is a 0 in the y column.

Find the intercepts and a 3rd point. Plot the points, make sure they line up, then draw the line.

The equation has only one variable.

x = a vertical.

y = b horizontal.

Key Concepts

Slope of a line between two points.
- The slope of the line between two points (x₁, y₁) and (x₂, y₂) is:
  
  m = $\displaystyle\frac{y_2 - y_1}{x_2 - x_1}$
How to graph a line given a point and the slope.
1. Plot the given point.
2. Use the slope formula m = $\displaystyle\frac{\text{rise}}{\text{run}}$ to identify the rise and the run.
3. Starting at the given point, count out the rise and run to mark the second point.
4. Connect the points with a line.

Slope Intercept Form of an Equation of a Line

- The slope–intercept form of an equation of a line with slope m and y-intercept, (0, b) is y = mx + b

Point-Plotting

Slope-Intercept

Intercepts

Recognize Vertical and Horizontal Lines

Create a blank table with two columns and four rows. The first row is labeled “x” and “y”. Find three points. Plot the points, make sure they line up, them draw the line.

y = mx + b

Find the slope and y-intercept, Start at the y-intercept, then count the slope to get a second point.

Create a blank table with two columns and four rows. The first row is labeled “x” and “y”. Plug a zero in the first row in the x column. In the second row there is a 0 in the y column.

Find the intercepts and a 3rd point. Plot the points, make sure they line up, then draw the line.

The equation has only one variable.

x = a vertical.

y = b horizontal.

We have seen that we can use either the slope-intercept form or the point-slope form to find an equation of a line. Which form we use will depend on the information we are given.

To Write an Equation of a Line
If given:	Use:	Form:
Slope and y-intercept	slope-intercept	y = mx + b
Slope and a point	point-slope	y − y₁= m(x − x₁)
Two points	point-slope	y − y₁= m(x − x₁)

Derived from Openstax Elementary Algebra, Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction. Intermediate Algebra, Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction. ↵

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Slope and Equations

Graphing and Slope

Graphing with Slope and Intercept

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