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Finding the Slope of a Line from 2 Points - Problem 1
Use the slope formula to find the slope of a line given the coordinates of two points on the line. The slope formula is m=(y2-y1)/(x2-x1), or the change in the y values over the change in the x values. The coordinates of the first point represent x1 and y1. The coordinates of the second points are x2, y2. It does not matter which point you label as the first point and which you label as the second. Don't forget to include the correct sign of each value. Simplify to get the value of the slope. Check the your answer by graphing the points and verifying that the vertical distance between the two points and the horizontal distance between the two points is captured by the numerator and denominator of the slope.
This is a problem that gives me two points and asks me to find the slope of the line that connects them. So if I wanted to, I could draw a graph, but to be honest with you guys, I don’t really like graphing. It takes a lot of time, I don’t always have graph paper, so this is an Algebra shortcut that I like to use, and that’s using the formula for slope. You guys should have this memorized, if you don’t yet, start memorizing.
M stands for slope, and you do y2 take away y1 on top of x2 take away x1. So in order to apply this formula, the first thing I need to do is figure out what this y2, y1 business means. I’m going to go ahead and label these points as x1 meaning my first x coordinate, y1 my first y coordinate, x2 is going to be 4 and y2 is going to be that 1.
Then I’m just going to substitute each one of these numbers into that formula. So let’s see y2 is 1 take away y1, there it is, that’s the top of my fraction. The bottom is going to be x2 take away x1. Oh my gosh please be careful with the negative sign. Notice how I had a negative, negative, one minus sign came from the equation and the other minus sign came from that 3 right there, that’s why I have minus, minus.
Okay well let’s simplify that. 1 take away 2 is -1, on the bottom I have 4 minus -3 which is the same thing as 4 plus 3 which is 7, that’s it. Using Algebra I was able to find the slope of the line containing those two points without having to graph it, I like that kind of shortcut. Before we move on, I want to show you guys just a quick little sketch of a graph that might help you to check your work. Like for example let’s just say I try graphing these points, I’m just going to verify that -1/7 makes sense.
Let’s see, so my first point is (-3,2), (-3,2) somewhere a round there, my second point is 4, 1 one, two, three, four, one and I’m looking at the line that passes through those. Again this is not very pretty because I’m not using graph paper, but I’m going down 1 box and over 7 boxes. If you guys had graph paper, you would have this a lot more precisely and you could see that indeed my slope is, oh my gosh sorry this is like the ugliest picture, but I hope you guys can still get the idea. My slope is down 1, over 7, that’s the negative represents the down, and then all over 7.
So the graphing is a good way to check, it can really rough like this just to make sure I didn’t accidentally put the Xs on top or something, and I didn’t accidentally make some weird calculation error, but one more thing I want to tell you guys before we go, is please remember this formula, it’s a great shortcut for how you can find the slope without having to graph a line.