How use the slope formula and find the slope of a line, whether the Slope is positive, negative or undefined. (2024)

Worksheet on Slope Of A Line

Slope Applet (html5)

Slope Formula Calculator(Free online tool calculates slope given 2 points)

The slope of a line characterizes the direction of a line. To find the slope, you divide the difference of the y-coordinates of 2 points on a line by the difference of the x-coordinates of those same 2 points.

How use the slope formula and find the slope of a line, whether the Slope is positive, negative or undefined. (1)

Different words, same formula

Teachers use different words for the y-coordinates and the the x-coordinates.

  1. Some call the y-coordinates the rise and the x-coordinates the run.
  2. Others prefer to use $$ \Delta $$ notation and call the y-coordinates $$ \Delta y$$ , and the x-coordinates the $$ \Delta x$$ .

How use the slope formula and find the slope of a line, whether the Slope is positive, negative or undefined. (2)

These words all mean the same thing, which is that the y values are on the top of the formula (numerator) and the x values are on the bottom of the formula (denominator)!

How use the slope formula and find the slope of a line, whether the Slope is positive, negative or undefined. (3)

Example One

The slope of a line going through the point (1, 2) and the point (4, 3) is $$ \frac{1}{3}$$.

Remember: difference in the y values goes in the numerator of formula, and the difference in the x values goes in denominator of the formula.

How use the slope formula and find the slope of a line, whether the Slope is positive, negative or undefined. (4)

Can either point be $$( x_1 , y_1 ) $$ ?

There is only one way to know!

First, we will use point (1, 2) as $$x_1, y_1$$, and as you can see : the slope is: $ \boxed {\frac{1}{3} }$ .

Now let's use point (4, 3) as $$x_1, y_1$$, and as you can see , the slope simplifies to the same value: $ \boxed {\frac{1}{3} }$ .

How use the slope formula and find the slope of a line, whether the Slope is positive, negative or undefined. (5)

The work , side by side

point (4, 3) as $$ (x_1, y_1 )$$

$$ slope = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{3-2}{4-1} = \frac{1}{3} $$

point (1, 2) as $$ (x_1, y_1 )$$

$$ slope = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{2-3}{1-4} = \frac{-1}{-3} = \frac{1}{3} $$

Answer: It does not matter which point you put first. You can start with (4, 3) or with (1, 2) and, either way, you end with the exact same number! $$ \frac{1}{3} $$

Example 2 of the Slope of A line

The slope of a line through the points (3, 4) and (5, 1) is $$- \frac{3}{2}$$ because every time that the line goes down by 3(the change in y or the rise) the line moves to the right (the run) by 2.

How use the slope formula and find the slope of a line, whether the Slope is positive, negative or undefined. (6)

This Page:

  • Formula
  • Example
  • Video
  • Order?
  • Do any two points determine the slope of a line?
  • Slope of Vertical a Line
  • Slope of Horizontal Line
  • Practice Problems

Video Tutorial on the Slope of a Line

Slope of vertical and horizontal lines

The slope of a vertical line is undefined

This is because any vertical line has a $$\Delta x$$ or "run" of zero. Whenever zero is the denominator of the fraction in this case of the fraction representing the slope of a line, the fraction is undefined. The picture below shows a vertical line (x = 1).

How use the slope formula and find the slope of a line, whether the Slope is positive, negative or undefined. (7)

The slope of a horizontal line is zero

This is because any horizontal line has a $$\Delta y$$ or "rise" of zero. Therefore, regardless of what the run is (provided its' not also zero!), the fraction representing slope has a zero in its numerator. Therefore, the slope must evaluate to zero. Below is a picture of a horizontal line -- you can see that it does not have any 'rise' to it.

How use the slope formula and find the slope of a line, whether the Slope is positive, negative or undefined. (8)

Do any two points on a line have the same slope?

Answer: Yes, and this is a fundamental point to remember about calculating slope.

Every line has a consistent slope. In other words, the slope of a line never changes. This fundamental idea means that you can choose any 2 points on a line.

How use the slope formula and find the slope of a line, whether the Slope is positive, negative or undefined. (9)

Think about the idea of a straight line. If the slope of a line changed, then it would be a zigzag line and not a straight line, as you can see in the picture above.

As you can see below, the slope is the same no matter which 2 points you chose.

The Slope of a Line Never Changes

How use the slope formula and find the slope of a line, whether the Slope is positive, negative or undefined. (10)

This Page:

  • Formula
  • Example
  • Video
  • Order?
  • Do any two points determine the slope of a line?
  • Slope of Vertical a Line
  • Slope of Horizontal Line
  • Practice Problems

Worksheet on Slope Of A Line

Slope Applet (html5)

Slope Formula Calculator(Free online tool calculates slope given 2 points)

Practice Problems

Problem 1

What is the slope of a line that goes through the points (10,3) and (7, 9)?

$\frac{rise}{run}= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Using $$ \red{ (10,3)}$$ as $$x_1, y_1$$

$\frac{9- \red 3}{7- \red{10}} \\ = \frac{6}{-3} \\= \boxed {-2 }$

Using $$ \red{ (7,9)} $$ as $$x_1, y_1$$

$ \frac{3- \red 9}{10- \red 7}\\ =\frac{-6}{3} \\= \boxed{-2 }$

Problem 2

A line passes through (4, -2) and (4, 3). What is its slope?

$\frac{rise}{run}= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Using $$ \red{ ( 4,3 )}$$ as $$x_1, y_1$$

$= \frac{-2 - \red 3}{4- \red 4} \\ =\frac{-5}{ \color{red}{0}}\\ = \text{undefined} $

Using $$ \red{ ( 4, -2 )}$$ as $$x_1, y_1$$

$= \frac{3- \red{-2}}{4- \red 4} \\ =\frac{5}{ \color{red}{0}} \\= \text{undefined} $

Whenever the run of a line is zero, the slope is undefined. This is because there is a zero in the denominator of the slope! Any the slope of any vertical line is undefined .

Problem 3

A line passes through (2, 10) and (8, 7). What is its slope?

$\frac{rise}{run}= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Using $$ \red{ ( 8, 7 )}$$ as $$x_1, y_1$$

$\frac{10 - \red 7}{2 - \red 8} \\ = \frac{3}{-6} \\ = -\frac{1}{2} $

Using $$ \red{ ( 2,10 )}$$ as $$x_1, y_1$$

$\frac{7 - \red {10}}{8- \red 2} \\ = \frac{-3}{6} \\ = -\frac{1}{2} $

Problem 4

A line passes through (7, 3) and (8, 5). What is its slope?

$\frac{rise}{run}= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Using $$ \red{ (7,3 )}$$ as $$x_1, y_1$$

$$\frac{ 5- \red 3}{8- \red 7}\\= \frac{2}{1} \\ = 2 $$

Using $$ \red{ ( 8,5 )}$$ as $$x_1, y_1$$

$$\frac{ 3- \red 5}{7- \red 8} \\= \frac{-2}{-1} \\ = 2 $$

Problem 5

A line passes through (12, 11) and (9, 5) . What is its slope?

$\frac{rise}{run}= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Using $$ \red{ ( 5, 9)}$$ as $$x_1, y_1$$

$$\frac{ 11 - \red 5}{12- \red 9}\\ = \frac{6}{3}\\ =2 $$

Using $$ \red{ (12, 11 )}$$ as $$x_1, y_1$$

$$ \frac{ 5- \red{ 11} }{9- \red { 12}}\\ = \frac{-6}{-3}\\ = 2 $$

Problem 6

What is the slope of a line that goes through (4, 2) and (4, 5)?

$\frac{rise}{run}= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Using $$ \red{ ( 4,5 )}$$ as $$x_1, y_1$$

$$\frac{ 2 - \red 5}{4- \red 4}\\ = \frac{ -3}{\color{red}{0}}\\ = undefined $$

Using $$ \red{ ( 4,2 )}$$ as $$x_1, y_1$$

$$\frac{ 5 - \red 2}{4- \red 4}\\ = \frac{ 3}{\color{red}{0}}\\ = undefined $$

WARNING! Can you catch the error in the following problem Jennifer was trying to find the slope that goes through the points $$(\color{blue}{1},\color{red}{3})$$ and $$ (\color{blue}{2}, \color{red}{6})$$ . She was having a bit of trouble applying the slope formula, tried to calculate slope 3 times, and she came up with 3 different answers. Can you determine the correct answer?

Challenge Problem

Find the slope of A line Given Two Points.

Attempt #1

$ slope= \frac{rise}{run} \\= \frac{\color{red}{y_{2}-y_{1}}}{\color{blue}{x_{2}-x_{1}}} \\= \frac{6-3}{1-2} \\= \frac{3}{-1} =\boxed{-3} $

Attempt #2

$$slope= \frac{rise}{run}\\= \frac{\color{red}{y_{2}-y_{1}}}{\color{blue}{x_{2}-x_{1}}}\\=\frac{6-3}{2-1}\\= \frac{3}{1} \\ = \boxed{3} $$

Attempt #3

$$ slope = \frac{rise}{run} \\= \frac{\color{red}{y_{2}-y_{1}}}{\color{blue}{x_{2}-x_{1}}} \\ =\frac{2-1}{6-3} \\ =\boxed{ \frac{1}{3}} $$

The correct answer is attempt #2.

In attempt #1, she did not consistently use the points. What she did, in attempt one, was :

$$ \frac{\color{red}{y{\boxed{_2}}-y_{1}}}{\color{blue}{x\boxed{_{1}}-x_{2}}} $$

The problem with attempt #3 was reversing the rise and run. She put the x values in the numerator( top) and the y values in the denominator which, of course, is the opposite!

$$ \cancel {\frac{\color{blue}{x_{2}-x_{1}}}{\color{red}{y_{2}-y_{1}}}} $$

Slope Practice Problem Generator

You can practice solving this sort of problem as much as you would like with the slope problem generator below.

It will randomly generate numbers and ask for the slope of the line through those two points. You can chose how large the numbers will be by adjusting the difficulty level.

Related pages:

Worksheet on Slope Of A Line

Slope Applet (html5)

Slope Formula Calculator(Free online tool calculates slope given 2 points)

How use the slope formula and find the slope of a line, whether the Slope  is positive, negative or undefined. (2024)

References

Top Articles
Latest Posts
Article information

Author: Prof. An Powlowski

Last Updated:

Views: 5553

Rating: 4.3 / 5 (44 voted)

Reviews: 83% of readers found this page helpful

Author information

Name: Prof. An Powlowski

Birthday: 1992-09-29

Address: Apt. 994 8891 Orval Hill, Brittnyburgh, AZ 41023-0398

Phone: +26417467956738

Job: District Marketing Strategist

Hobby: Embroidery, Bodybuilding, Motor sports, Amateur radio, Wood carving, Whittling, Air sports

Introduction: My name is Prof. An Powlowski, I am a charming, helpful, attractive, good, graceful, thoughtful, vast person who loves writing and wants to share my knowledge and understanding with you.