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Determining Slope

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Today we're going to talk about determining the slope of a line. Remember, the slope of a line just tells you how steep it is. So we're going to start by looking at the definition of slope, and then we'll do some examples, finding the slope both from a set of points, and from a table of values.

So let's start by looking a little bit more closely at what slope means. So again, slope you can think of as how steep a line is. And so we're comparing a vertical distance between two points on a line to the horizontal distance between two points on a line. So when we're calculating slope, we want to find the change or the difference in their y-coordinates, and then we're going to divide that by the change or difference in the x-coordinates.

So when we're calculating slope using a formula, we usually use m to stand for slope, and we have the difference of our y-coordinates, Y2 minus Y1 in the numerator of a fraction, and the difference of our x-coordinates, x2 and x1, in the denominator of our fraction.

And so from this formula we can see that if we know the coordinates, or the x and y values of any two points on a line, we can then go ahead and calculate the slope. So let's do some examples using this formula. So here's my first example. I've got a line, on our coordinate plane, and I want to use my slope formula to calculate the slope of that line. So to do that, I need to know two coordinate points on my line, so that I can find the x and y values to substitute into my formula.

So I'm going to-- I can pick any two points on this line. I'm going to pick this first point, which has an x value of negative 5, and a y value of negative 3. And then I also use this point which has an x value of negative 1, and a y value of 0.

So I'm going to label this point as x1, and y1. And I'll call this point x2 and y2. So now that I have my values, I can go ahead and substitute them into my formula to find the slope.

So to do that, in my slope formula, in the numerator, I have my Y2 and my Y1 values. So I'm going to substitute 0 and negative 3 into the numerator. So I have 0 minus negative 3. The numerator of my fraction, and then I want to subtract my x values.

So I'm going to be subtracting negative 1 and negative 5. So I'll have negative 1 minus negative 5. Simplifying this, 0 minus negative 3, gives me a positive 3. And in the denominator, negative 1 minus negative 5 is going to give me 4.

So I found that using my formula, The slope of this line is three fourths, for 3 over 4. So we can verify that by looking at our line, and counting to find the rise in the run between pairs of points. So if I start with this point, the rise, remember, is our vertical distance. So if I count up one two three, I see that the rise is three.

And if I count horizontally, to the right, "1, 2, 3, 4," I see that the run is four. So the rise over run, or the slope is 3 over 4. So that matches up with what we found with our formula. I can see doing that between another two pairs of points that the rise over run should also be 3 over 4.

So counting up one two three and counting over one two three four OK I've verified that my rise over run is also 3 over 4 between these two pairs of points so let's do another example.

So let's look at how we can calculate the slope of a line, given an equation, by creating a table of x and y values. So this equation, if we were to graph it on a coordinate plane, would give us a line similar to the one that we saw in our last example. And so if we want to calculate the slope, we can create a table of values by substituting some x values into our equation, to figure out the y values that correspond with each of those x values.

So if I start with the x value of negative 2, I'm going to substitute that into my equation for x. 2 times negative 2 is going to give me negative 4. If I add one to that, that will give me a negative 3. So I found that my y value is negative 3 when I substitute in negative 2 for my x value. Let's try when x is negative 1. 2 times negative 1 is going to give me a negative 2. Add one to that, and that will give me negative 1.

So I found that my y value is negative 1, again, when x is negative 1. Substituting 0 into our equation for x, 2 times 0 is 0, plus 1, will give me a wide value of one. Substituting one into our equation for x, 2 times 1 is 2, plus 1 will give me three. And finally, substituting 2 into our equation for x, 2 times 2 is 4, plus 1, will give me a y value of 5.

So now that I have the table of values, I can use my formula for slope to find the difference in y values and the difference in x values for any two pair of points within this table. So when you are calculating the slope, you only need to look at two pairs of points. And you can choose any two pairs of points on that line to calculate the slope.

So I'm going to start by choosing these last two pairs of points. So thinking about my formula for Slope, I know that I'm going to find the difference in my y values, in the numerator. So 5 minus 3. And then I'm going to find the difference in my x values. So 2 minus 1. Simplifying this, 5 minus 3, will give me 2. 2 minus 1 will give me 1, and 2 over 1 is the same as 2. So I found that the slope of this line is going to be two. Let's verify that using just another pair of x and y values.

So let's try 0, 1, and negative 1, negative 1. So again I'll have the difference in my y values-- so 1 minus negative 1. And then, in the denominator, the difference in my x values. So 0 minus negative 1. Simplifying this, one minus negative 1, would give me a positive 2, and 0 minus negative 1 will give me a positive 1. Again, 2 over 1 is just 2. So we found that between these two pairs of points our slope is also two, which makes sense, because between any two pairs of points on a line your slope is going to be the same.

So you may have noticed that the slope that we found is in our equation. And this is not a coincidence. When you have an equation of a line in this form, the slope of the line is going to be the coefficient of the x term. So when you have an equation that looks like this in this form, you can simply look at the coefficient of the x term to see the slope, instead of using the formula.

So let's go over our key points from today. Make sure you have them in your notes if you don't already, so you can refer to them later. The slope of a line is commonly referred to as the rise over run, with the rise is the difference in y-coordinates, and the run is the difference in x-coordinates between any two points on a line. And to determine the slope, you only need the x and y-coordinates of two points on a Line.

So I hope that these key points and examples helped you understand a little bit more about determining slope. Keep using your notes, and keep on practicing. And soon you'll be a pro. Thanks for watching.

Formulas to Know

- Slope