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NY Regents Exam - Integrated Algebra: Help and Review25 chapters | 272 lessons | 12 flashcard sets

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we explore the definition of collinear points and how to recognize them in our environment. We learn how to determine if a given set of points are collinear by exploring graphs and slopes.

It is incredibly common to look up and see a bunch of birds lined up, perched on a group of power lines. Have you ever wondered how they do this without getting shocked? The fact is that birds don't get shocked while sitting on the power lines, because they are not good conductors of electricity compared to the wire. For this reason, the electricity passes right by them and continues along the wire.

Another interesting fact about seeing birds on these power lines is that the set of birds on each individual power line can be said to be collinear. In mathematics, a set of points that all lie on the same line are **collinear**. When you see birds all lined up along a power line, you see that they all lie on the same line. Another example of collinear points would be musical notes on sheet music. We see that musical notes of the same key lie on the same barline, making them collinear.

Mathematically speaking, it's easy to see that any two points are collinear. This is because you can draw a line through any two points, so they must lie on the same line. But how do we tell if a set of points are collinear if there are more than two of them? For instance, if I gave you the points (0,3), (4,5), and (-2, 2), how can we determine if they all lie on the same line? One way to do this would be to graph them and see if they all lie on the same line.

In this example, it appears that the three points are collinear, and in fact, they are. However, the downside to this is that you can sometimes get a point that is very close to a line and looks like it lies on it, but it actually doesn't. For instance, in our example, the three points all lie on the line *y* = ½*x* + 3. What if I added the point (2, 3.9) to the set, and I wanted to know if the points (0,3), (4,5), (-2,2), and (2, 3.9) were all collinear. If we plotted this extra point on the graph, it would appear that it fell on this line.

However, it actually doesn't, because it doesn't satisfy the equation of the line. If we plug in 2 for *x*, then our *y* ends up being (1/2) * 2 + 3 = 4, not 3.9. Thus, in reality, the point doesn't lie on the same line, so the set of those four points would not be collinear. So, how can we tell for sure? The answer lies in the slope of a line. A line only has one slope, so if any set of points all lie on the same line, then the slope of the line between each pair of points in that set must be the same.

**Slope** is the rate at which *y* is changing with respect to *x* along a line. We calculate the slope of a line through two points (*x*1, *x*2) and (*y*1, *y*2), by finding the change in *y*-values and dividing by the change in *x*-values. In other words, slope = (*y*2 - *y*1) / (*x*2 - *x*1). For example, the slope of the line through (3,4) and (7,10) is (10 - 4) / (7 - 3) = 6/4 = 3/2.

If we want to determine if a set of points are collinear, we just calculate the slope of the lines through each of the pairs of points. If they are all the same, then they all fall on a line with the same slope, so they all fall on the same line; that is to say, they are collinear. If the slopes are different between any two pairs, then the points do not fall on the same line and they are not collinear.

For example, when we look at our example of the three points (0,3), (4,5), and (-2,2), we calculate the slope between each pair of points.

Between the points (0,3) and (4,5), we calculate the slope as follows: (5 - 3) / (4 - 0) = 2/4 = ½. Let's do the same for the points (0,3) and (-2,2). The slope is equal to (2 - 3) / (-2 - 0) = -1/-2 = ½. Finally, we have the points (4,5) and (-2,2). Slope equals (2 - 5) / (-2 - 4) = -3/-6 = ½.

We see that the slope of the line between each pair of points is the same, thus the points are collinear. Now, consider if we had the four points (0,3), (4,5), (-2,2), and (2,3.9), where we added in the (2,3.9) like we said earlier. We know that the first three points all lie on the same line, so we just need to calculate the slope of the lines between the new point and first three points to see if the slope is the same.

First, let's find the slope between (0,3) and (2,3.9). Slope = (3.9 - 3) / (2 - 0) = 0.9/2 = 9/20. We can stop there. We see that the slope of the line between (0,3) and (2,3.9) is 9/20 which is not equal to ½, so (2,3.9) can't lie on the same line as (0,3), (4,5), and (-2,2).

It's easy to spot objects that appear to be collinear in the environment around us. Any objects that lie on the same line are **collinear**. Mathematically, if we are given a set of points and we want to determine if they are collinear, the best way to do this is by calculating the slope of the line between each pair of points in the set to see if they are equal. If they are, then the points are collinear. If not, then they are not collinear. We now have the information we need to be comfortable with the definition of collinear points and to determine if the given points in a set are collinear.

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NY Regents Exam - Integrated Algebra: Help and Review25 chapters | 272 lessons | 12 flashcard sets

- What Are the Different Parts of a Graph? 6:21
- Graph Functions by Plotting Points 8:04
- Identify Where a Function is Linear, Increasing or Decreasing, Positive or Negative 5:49
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