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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[2]

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 the 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
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 slope first and then use one of the points.

  1. Find the slope using the given points. m = \displaystyle\frac{y_2 - y_1}{x_2 - x_1}
  2. Choose one point.
  3. Substitute the values into the point-slope form: y − y1 = m (x − x1)
  4. 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:

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

m = \displaystyle\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 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:

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

Find the slope of the line through(−3,5)and(−3,4). 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:


Step 1 is to find the slope using the given points. Find the slope of the line through (negative 3, negative 1) and (2, and negative 2). m equals the quotient of y 2 minus y 1 in parentheses and x 2 minus x 1 in parentheses. m equals the quotient of negative 2 minus negative 1 in parentheses and 2 minus negative 3 in parentheses. m equals negative 1 divided by 5.Step 2 is to identify the point. Choose either point. x 1 is 2 and y 1 is negative 2.Step 3 is to substitute the values into the point-slope form y minus y 1 equals m times the quantity x minus x 1 in parentheses. y minus negative 2 equals negative 1 divided by 5 times the quantity x minus 2 in parentheses. This simplifies to y plus 2 equals negative 1 divided by 5 times x plus 2 divided by 5.Step 4 is to write the equation in slope-intercept form. y equals negative 1 divided by 5 times x minus 8 divided by 5.

 

In the examples above you will note 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

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:

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.

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

Solution:

y = –\displaystyle\frac{2}{5}x – 1

#3 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:

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.
Find the slope, by counting the rise and run. m equals rise divided by run.

m equals 2 divided by 3.
Find the y-intercept.
Find the y-intercept. y-intercept is (0, negative 4).
Substitute the values into y=mx+b
Substitute the values into y equals m x plus b.

y equals 2 divided by 3 x minus 4. The m and 2 divided by 3 are both emphasized in red. The b and negative 4 are emphasized in blue.

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

Solution:

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.
Identify the slope. m equals 0.
Identify the point.
Identify the point. x 1 is negative 2 and y 1 is negative 6.
Substitute the values into

y−y1=m(x−x1)
Substitute the values into y minus y 1 equals m times the quantity x minus x 1 in parentheses.

y minus negative 6 equals 0 times the quantity x minus negative 2 in parentheses.
Simplify.
This simplifies to y plus 6 equals 0 or y equals negative 6.

Write in slope-intercept form. It is in y-form, but could be written y equals 0 x minus 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[3]

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}.

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.Step 2 is to use the slope formula m equals rise divided by run to identify the rise and the run. Identify the rise and the run. m equals 3 divided by 4. Rise divided by run equals 3 divided by 4. Rise equals 3. Run equals 4.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.Step 4 is to connect the points with a line. Connect the two points with a line. . The figure then shows the graph of a straight line, three points, and two 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). A horizontal line segment connects (1, 2) to (5, 2) and is labeled 4. A straight line is drawn through the points (1, negative 1) and (5, 2) with arrows on both ends.

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:

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:

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.
  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.

Graphing with Slope and Intercept[4]

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 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, 3), (2, 4), and (4, 5). A right triangle is drawn by connecting the three points (2, 4), (2, 5), and (4, 5). The vertical side of the triangle is labeled “Rise equals 1”. The horizontal side of the triangle is labeled “Run equals 2”. The line is labeled y equals 1 divided by 2 x plus 3.

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

m = \displaystyle\frac{\text{rise}}{\text{run}}
m = \displaystyle\frac{1}{2}

The y-intercept is (0,3).

Look at the equation of this line.

The figure shows the equation y equals 1 divided by 2 x plus 3. The 1 divided by 2 is emphasized in red. The 3 is emphasized in blue.

 

 

Look at the slope and y-intercept.

Slope m equals 1 divided by 2 and y-intercept (0, 3). The 1 divided by 2 is emphasized in red. The 3 is emphasized in blue.

 

 

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.”

m equals 1 divided by 2; y-intercept is (0, 3). y equals 1 divided by 2 x plus 3. y equals m x plus b. The m and 1 divided by 2 are emphasized in red. The b and 3 are emphasized in blue.

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. 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. y=mx+b
Write the equation of the line. y= 4/7x-2
Identify the slope. m= - 4/7
Identify the y-intercept. Y-intercept is (0,-2)

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. x + 3y = 9
Subtract x from each side. 3y=-x+9
Divide both sides by 3. 3y/3 = (-x+9)/3
Simplify. y= -1/3x + 3
Write the slope–intercept form of the equation of the line. y-mx+b
Write the equation of the line. y=- 1/3x +3
Identify the slope. m= -1/3
Identify the y-intercept. y-intercept is (0,3)

 

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:

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:

Steps Algebraic
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: 

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.

The table has a title row that reads “Methods to Graph Lines”. Below this are four columns. The first column contains the following: Point Plotting. A blank table with two columns and four rows. The first row is a header row with the headers “x” and “y”. Find three points. Plot the points, make sure they line up, them draw the line. The second column contains: Slope-Intercept. Y equals m x plus b. Find the slope and y-intercept, then count the slope to get a second point. The third column: Intercepts. A table with two columns and four rows. The first row is a header row with the headers “x” and “y”. In the first row there is a 0 in the x column. In the second row there is a 0 in the y column. The remaining spaces are blank. Fourth column. Recognize vertical and horizontal lines. The equation has only one variable. X equals a vertical. Y equals 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
    • The table has a title row that reads “Methods to Graph Lines”. Below this are four columns. The first column contains the following: Point Plotting. A blank table with two columns and four rows. The first row is a header row with the headers “x” and “y”. Find three points. Plot the points, make sure they line up, them draw the line. The second column contains: Slope-Intercept. Y equals m x plus b. Find the slope and y-intercept, then count the slope to get a second point. The third column: Intercepts. A table with two columns and four rows. The first row is a header row with the headers “x” and “y”. In the first row there is a 0 in the x column. In the second row there is a 0 in the y column. The remaining spaces are blank. Fourth column. Recognize vertical and horizontal lines. The equation has only one variable. X equals a vertical. Y equals 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)

 

  1. Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction ↵
  2. Section partially derived from Openstax Intermediate Algebra: Find the Equation of a line ↵
  3. Section derived from Openstax Intermediate Algebra: Slope of a Line ↵
  4. Section derived from Openstax Intermediate Algebra: Slope of a Line ↵

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