Does this point lie on this line? The Point-Slope format (y – y 1 ) = m(x – x 1 )

The Algebra Standard California Algebra 1 content standard #7 –Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.

Am I ready? Before watching this video you should know. –What the X-Y plane is. –How to plot points on the plane. –How to read the co-ordinates of a point drawn on an X-Y plane. –You can work math quickly, multiplying, adding, subtracting, including positive and negative numbers.

A Line is a Set of Points In Algebra a line is created by connecting together an infinite set of points. These points are GENERATED by working a particular math equation over and over again with different inputs. This relationship between a line and an equation can be quite difficult for students to get a first, but once you get it a whole new world of mathematics is opened up to you. This fundamental skill is used all the way through Pre-Calculus and Calculus.

y=2x + 3 X is used as INPUT Y is used as OUTPUT Each Y is Generated by working the math after we plug in a different value for X. We choose our Xes. They can be anything we want, but the Ys must come from the math. Let’s look at this equation:

An equation can take many forms. Keep in mind that an equation can look a lot of different ways but the important thing that they all have in common is they all have and input variable and an output variable. In math we also call these the INDEPENDENT variable, and the DEPENDENT variable.

The many forms I will show you what I mean: y=2x + 3 y + 1 = 2(x + 5) y – 2x = 3 y – 3 = 2x ALL of these say fundamentally the same thing. For any x filled into any one of these the same y will be generated.

x y Let put these points onto a graph y=2x + 3

It’s about TRUTH All of math is about truth. How do you tell the difference between a point the lies on the line and a point the is not on a line? It’s easy. An X and Y replacement that creates a true statement is on the line. An X and Y replacement that creates a false statement is NOT on the line.

Be flexible The line can move. The equation can change form. The point can move. The only thing that you can count on is, working the math and looking for the TRUTH.

OK, time to do some real examples. Example 1: Does the point (3,8) lie on the line: y = 3x -1? Simple process. –Plug in the x. –Plug in the y. –See if you get a simple true statement.

(3,8) in y = 3x -1? Recall that (3,8) is an ordered pair. Just from their position you should know that x = 3 and y = 8. Because an ordered pair always occurs in the same order: (x,y)

So we have: An equation: y = 3x -1 and we have x = 3 and y = 8 Carefully fill these values into the right place. Don’t forget to GET RID of the letter when you put in the number. (Y) = 3 ( X ) – 1 (8) = 3 ( 3 ) – 1

Move to truth, we hope. (8) = 3 ( 3 ) – 1 8 = 9 – 1 8 = 8 TRUE!! This is true so, the point DOES lie on this line.

Another example Let’s look at another example: Does the point (2,-3) Lie on the line y = -4x + 5? Once again we will follow the simple process: –Plug in the X. –Plug in the Y. –Do the math and see if you get truth?

y = -4x + 5 Is (2,1) in this point in this linear equation? What is X? 2 What is Y? 1 Plug in these numbers: (1) = -4(2) + 5

Carry this math through (1) = -4(2) = = -3 This is false so, the point does NOT lie on this line.

First part done The first part of the standard is now done. “Students verify that a point lies on a line, given an equation of the line.” You should now know how to tell if a point lies in an equation or not.

Review Let’s review this simple process: Plug in the x and y of a point that you would like to check. Do the math until you get a very simple statement: If the statement is true then the point IS on the line. If the statement is false then the point is NOT on the line.

Part 1 done Continue to part II to fully cover the standard and learn how to use the Point-Slope form of a linear equation. (y + y 1 ) = m(x +x 1 )

Part 2 The Point-Slope Form of a linear equation looks like this: (y + y 1 ) = m(x + x 1 ) This says in English: “An output variable, called y when added to a sample y, called y (sub one) should be equal to the slope of the line (m) multiplied by the quantity of the the input variable (x) added to a sample x, called x (sub one).

Why the name? The Point-Slope form is called the point slope form because, in order to work with it all that you need is a single point and the slope of the line. OR if you have any two points that lie on the line you can calculate your slope. We will show you how.

Example 1 If a line contains the point (2, 5) on a line and the slope is -1/2. What does the equation look like in Point-Slope form? (y + y 1 ) = m(x +x 1 ) y 1 is the y value of our given point. x 1 is the x value of our given point. m is the slope.

Simple So you see this is a simple process. Just be CAREFUL. Plug the right number into exactly the right place. These boxes are the ones that get replaced. The x and the y are un-touched. (2,5) slope = - 1/2 (y + ) = (x + )

Final answer: (y + 5) = -1/2 (x + 2) THIS is the answer that the example is looking for. It is not a number. It is an equation. Remember, in Algebra, you have to read the instructions carefully and provide exactly the type of answer that the problem is looking for.

Algebra instructions are hard. Be flexible, ready to handle anything. Remember, in Algebra, you have to read the instructions carefully and provide exactly the type of answer that the problem is looking for. It is often not just a number. You may need to produce: –An equation. –A set of two numbers. –A precise drawing. –An inequality. –A set of more than two numbers.

Don’t expect to get this right away It will take practice before you can memorize this form and how to use it. Do lots of homework problems and it will slowly come.

Now let us fully live up to the standard For a straight line, all you need are two points to define it and you can create an equation that can generate thousands of points that lie on the same line. But you have to use the Point-Slope form, and the Slope formula to do this.

The Slope Formula To know the slope of anything you need to how fast it goes up per how far it goes along. To measure the up and down on the X-Y plane you use Y. To measure the left to right change you use X.

Let’s look at this line: Picking two points on the line we can go to the X and Y axies and get the co-ordinates. Then from these we can get the slope. (6,2) (4,-2)

The slope formula Here we are taking two ys and subtracting one from the other to get change in Y, the RISE. We are then dividing by difference between the two xs to get the RUN of the line.

(6,2) (4,-2) Identify: –x 1 = 6 –y 1 = 2 –x 2 = 4 –y 2 = -2

Slope formula filled in: -22 (6,2) (4,-2) 46 m = -4 -2

Simplify -4 / -2 = -2 / -1 The negatives cancel and we don’t have to state the one so… The slope is: 2

We are not quite done Back to the Point-Slope form: We have just created an m, a slope, that we can plug into this formula. We found that 2 is our slope but what else do we need? We need a y 1 and an x 1. Remember and be careful. We do NOT make a substitution for the X or the Y. Only the y (sub one) and the x (sub one). (y + y 1 ) = m(x +x 1 )

Where to get the subs? These numbers are meant to come from any single point that lies on the line. Therefore either of our two points will do. Let’s take: (6,2) With this point as our sample: –x 1 = 6 –y 1 = 2

Plug all this in and we are done: X sub one = 6 Y sub one = 2 m (our slope) = 2 The formula says: (y + y 1 ) = m(x +x 1 ) (y + 2) = 2(x + 6)

This Lesson is DONE. But you are not done. Memorize these two formulas and their names: The SLOPE formula. The Point-Slope form (y + y 1 ) = m(x +x 1 )

Learn to use them Learn what places to fill in. What places to leave alone. Learn how to generate what these formulas need, either from two simple points, or from a given point and a given slope. (y + y 1 ) = m(x +x 1 )

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