Computing the Slope of a Line 05:54 minutes

Video Transcript

Transcript Computing the Slope of a Line

Grandpa Lindbergh is one of the most famous aviators ever. He's always on the look-out for things to buy for his favorite plane...

In the plane's cockpit, Grandpa just installed some of the latest and greatest technology that money can buy. Today, he's gonna finally test it out and see how steep he can fly the plane. To do this, Grandpa Lindbergh will need help computing the slope of a line. Up, up, and away!

Looking at Grandpa’s flight path we can identify the coordinates of at least three points on the graph. Let’s take a look: To calculate the steepness of the flight path, we need to figure out the slope of the line. We can do this by using any two points on the line and plugging them into the slope formula. The slope formula is 'm' is equal to delta 'y' over delta 'x'. Delta is mathematical shorthand for change, or difference. That means it's the change in 'y' divided by the change in 'x'. The result, 'm', measures the steepness of the line connecting the two points.

Let’s plug in the coordinates for point 'a' and point 'b' into the slope formula. It doesn't matter which ordered pair you put first and which you put second just be consistent in how you order the 'x' and 'y' coordinates. Ok, we're going to use point 'a' as the first ordered pair and point 'b' as the second ordered pair. We plug in 3 for 'y' 2, 0 for 'y1', 2 for 'x2', and 0 for 'x1' which gives us the quantity '3 minus 0' divided by the quantity '2 minus 0' which is 3 over 2. This means for every increase of 2 unit on the x-axis there is an increase of 3 units on the y-axis.

A quick way to determine the slope is to draw a slope triangle on the line. Using line segment AB as the hypotenuse draw in a right triangle below the line. Remember, the hypotenuse is the longest side of a right triangle. It's also the side that's opposite the right angle. Just as we saw before, the change in 'y' is equal to 3 and the change in 'x' is 2. This is the same as what we calculated with the slope formula since it equals 3 over 2.

Let’s try two different points on the same line. This time, we’ll plug the coordinates for point 'a' and point 'c' into the formula. Once again, 'x1' is 0 and 'y1' is 0 but this time, 'x2' is 4 and 'y2' is 6. Subbing these values in gives us the quantity '6 minus 0' divided by the quantity '4 minus 0'. Simplifying like before, 'm' is equal to 3 over 2. How about that! The slope is the same as before. Don’t forget to draw in the slope triangle. Here, we can see that the change in 'y' is 6, and the change in 'x' is 4 and, 6 divided by 4 gives us, once again 3 over 2.

Let’s try it a third time but this time, we won’t use the ordered pair of point 'a' - just to check if that makes a difference. For point 'b' and point 'c', we again use the slope formula. Since we've already done it twice, let's go a bit more quickly. Once again, 'm' is equal to 3 over 2. And from the slope triangle we can see the slope is the same for these points too!

Let's summarize what we've learned. We know that the slope of a line can be calculated by plugging any two points of the line into the slope formula. We also know that the slope of a line is the same for ANY two points along that line. We can choose 'a' and 'b', 'b' and 'c', or 'a' and 'c'. We can extend what we've learned to write a rule about slope triangles. We learned that slope triangles created on one line, no matter what two points are used, will be similar triangles. Similar triangles share the same ratio between their sides, and always have the same angle measures.

Grandpa is really excited because everything seems to be working... Whoops! Wrong button...maybe he got a little too excited.