Back To Course

Math 101: College Algebra12 chapters | 94 lessons | 11 flashcard sets

Watch short & fun videos
**Start Your Free Trial Today**

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 55,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Jeff Calareso*

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Simply knowing how to take a linear equation and graph it is only half of the battle. You should also be able to come up with the equation if you're given the right information.

Mathematicians like to be thorough and find all the connections between things. So just being able to go from an algebraic rule to drawing the graph isn't really good enough for them. They want to be able to go the other way, too. Going from the graph to the rule isn't too bad because you can just pick out the information you need. But sometimes the picture isn't always going to be given and only small pieces of information will be given to you, and it's up to you to fill in the gaps.

There's a lot of different kinds of information that could be given. The most common is that you'll be given two points. Maybe they'll ask you to find the equation of the line between the points (3,4) and (6,-2).

For any linear equation, the two things we need to know are the **slope** and the **y-intercept**. The slope is every time going to be the first thing we should try to find. Once we know the slope, it's a lot easier to find the y-intercept.

I personally like to find the slope when I'm given two points just using the logic that I know because I know the slope is the rise over the run, which means how much it goes up and down over how much it goes left and right. I know the rise is the change in y. So in this one, I see the y starting at 4 and going down to -2, which means the rise is -6. The run, the left and right, is x, and I see it start at 3 and go up to 6, which is a change of 3. And so I can get the slope right from this, which is -2.

A lot of people like using the formula, which is the x1, y1, x2, y2 formula. If you do it this way, you'll get the exact same answer. You plug in numbers y2 minus y1 over x2 minus x1 (y2 - y1 / x2 - x1). You do -2 -and get -6. And 6 - 3 = 3. And again, we end up with a slope of -2.

Like I said earlier, the two things we need are the slope and the y-intercept. We now know what the slope is, which means the y-intercept is the only thing left to find. Because I know m (-2) and because it gave us a sample x and y (3,4), I can substitute in everything I know and I'm left with an equation with only one variable in it (4 = -2 * 3 + b).

First, I do the operation it asks us to: -2 * 3 = -6. Then, I have to get the b by itself, which means I undo the -6. I undo subtraction with addition because of inverse operations. I end up with b = 10. Now that I know m and I know b, I'm done, and my answer is y = -2x + 10. That means that this linear equation begins at 10 and is decreasing by 2 every step of the way.

So that' s just one example of some information you may be given. But there's a lot of different things that could be given to you and we've only touched on one of those things.

Another very common thing to see is that instead of you getting two points, you only get one point, but they tell you that your line is either parallel or perpendicular to other sample line that they tell you. In order to solve this question, you have to know what the deal with parallel and perpendicular lines is. **Parallel lines** are two lines that kind of look like train tracks. They go in the same direction that ends up meaning that they have the exact same slope. They go over and up the exact same amount. So parallel lines have the same slopes.

**Perpendicular lines**, on the other hand, are two lines that intersect each other at right angles. Perpendicular lines' slopes are what we call **opposite reciprocals**. One line is going over a lot, and up a little and one line is going over a little and down a lot. That has to do with the reciprocal part, which essentially means that if you have a fraction (1/4), you flip it (4/1).

So opposite reciprocals are numbers where one is positive (1/4) and one is negative (-4/1), and one is one fraction and the other is the flipped fraction. Don't forget to do both. It's really common to just make it negative or just make it positive, or forget to flip it or flip it but forget to change the sign. It's both.

Now that we know about parallel and perpendicular lines, we can answer questions like this that ask us, 'What is the equation of a line that is parallel to y = 3x + 2 and through the point (-3,6).'

Again, the two things we need to find are m and b, the slope and the y-intercept. We always want to find the slope first, so now I have to go about finding the slope in a slightly different manner than we did earlier. I can no longer use the slope formula, because it only gives us one point, so I have to use what I know about parallel lines.

Luckily, we just learned that parallel lines have the same slope, which means that my slope is the exact same as this one, so I can just take the slope from this line (3) and just steal it and I already know the slope; all my work is done and I have half the answer to my question (y = 3x + b)

The other half that I still need is to find the b and I'm going to do this in the exact same way that we just did. I now know a sample x (-3), a sample y (6), I know what m is (3), all I have to do is find b. I can substitute in what I know (6 = 3 * -3 + b). I can solve the equation for b by doing inverse operations to get the b by itself, and we find that b = 15. Now that we know b and m, we have our answer, which is y = 3x + 15.

Similarly, you could be asked to find the equation of the line that is perpendicular to this one through this point. We've got the same equation (y = 3x + 2) and the same point (-3,6), but now instead of it being parallel, it's perpendicular.

I still need to find the m and the b, just like before, and again, I cannot use the slope formula because I only have one point. That means we use what we know about perpendicular lines and how their slopes relate. I know that the slope of y = 3x + 2 is 3, but because it's perpendicular, I can't simply steal that.

I have to first take the opposite reciprocal of it. The opposite part means that it changes from positive (3) to negative (-3). The reciprocal part means that it changes from -3/1 to -1/3. And now, we have the slope (-1/3) of our line, which means that we have half our answer (y = -(1/3)x + b) and all that's left to find is b.

We find b in the exact same way we found b in every other problem in this video. We substitute in our sample x (-3), our sample y (6), what we know m is (-1/3) and we solve the equation for b (6 = -(1/3)(-3) + b). I multiply what I can and I undo to get b by itself, and this time we find that b = 5. This means my solution is y = -(1/3)x + 5.

To review, we learned that for any equation, we need to know what is the slope and what is the y-intercept. Once we know those two things, we're done and we can substitute them into **y = mx + b**.

The way we actually find m and b differs depending on the information that's given. We like using the **slope formula** (y2 - y11 / x2 - x1) if it gives us two points. Or, we might have to use what we know about **parallel and perpendicular lines**. We always find m first and then substitute in to find b.

You need to remember that parallel lines have slopes that are the same, whereas perpendicular lines have slopes that are **opposite reciprocals**.

To unlock this lesson you must be a Study.com Member.

Create
your account

Already a member? Log In

BackDid you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
7 in chapter 1 of the course:

Back To Course

Math 101: College Algebra12 chapters | 94 lessons | 11 flashcard sets

- What are the Different Types of Numbers? 6:56
- What Are the Different Parts of a Graph? 6:21
- What is a Linear Equation? 7:28
- Linear Equations: Intercepts, Standard Form and Graphing 6:38
- Abstract Algebraic Examples and Going from a Graph to a Rule 10:37
- Graphing Undefined Slope, Zero Slope and More 4:23
- How to Write a Linear Equation 8:58
- How Do I Use a System of Equations? 9:47
- Go to Foundations of Linear Equations

- Go to Functions

- Pennsylvania Grades 4-8 Core Assessment - Mathematics & Science (5155): Study Guide & Test Prep
- Literary Elements Lesson Plans & Resources
- Pennsylvania Grades 4-8 Core Assessment - English Language Arts & Social Studies (5154): Study Guide & Test Prep
- Space Science Lesson Plans & Activities
- Certified Safety Professional (CSP): Exam Prep & Study Guide
- Required Assignments for Business 302
- U.S. Colonial Resistance Lesson Plans
- Common Core Lesson Plan Templates & Resources
- Team Building Lesson Plans & Resources
- Settings in Literature Lesson Plans & Resources
- California Code of Regulations for Schools
- WV Next Generation Standards for Math
- Continuing Education Opportunities for Microbiology Technologists
- Professional Publications in Literacy
- Dyslexia Programs in Texas
- Study.com's Teacher Edition
- Study.com School Plans

- Analogies Lesson for Kids: Definition & Examples
- What is an Interior Designer?
- The Adventure of the Dying Detective Story Summary
- Verbal Learning: Methods, Types & Processes
- An Angel in Disguise: Summary & Quotes
- Goals of Conflict Management Programs in Schools
- Conversation Starters For Kids With Asperger's
- Sea Cucumber Lesson for Kids: Definition & Facts
- Quiz & Worksheet - Outlook Post-Basal Ganglia Stroke
- Quiz & Worksheet - Opposite Numbers
- Quiz & Worksheet - Political Factors in Economic Development
- Quiz & Worksheet - What is Tourism?
- Quiz & Worksheet - What is Apiculture?
- Graphs & Charts in Everyday Life Flashcards
- Interpreting & Analyzing Data Sets Flashcards

- Internet & Social Media Marketing: Help & Review
- MTEL English: Practice & Study Guide
- 8th Grade Life Science: Enrichment Program
- ORELA Mathematics: Practice & Study Guide
- Money Management: Help & Review
- The Role of Agency in Business Law
- The Great Depression (1929-1940)
- Quiz & Worksheet - Felt Emotions vs. Displayed Emotions
- Quiz & Worksheet - What is Utility Theory?
- Quiz & Worksheet - Coercive Power in Leadership
- Quiz & Worksheet - Benefits of Multitasking
- Quiz & Worksheet - Plain Folks Appeal in Advertising

- Set Notation: Definition & Examples
- Facts About Saturn: Rings, Temperature & Size
- Addition Math Games
- Science Word Walls
- Difference Between the ASVAB & AFQT
- Chinese New Year Lesson Plan
- Following Directions Activities & Games
- 5th Grade Math Centers
- 3rd Grade Math Centers
- Life Cycle of a Frog Lesson Plan
- Grants for Field Trips
- Community College Teaching Jobs

Browse by subject