Slope
Slope, the single most important idea to understand about lines in the x-y plane is the idea of a slope. Slope is, roughly, a measure of how steep a line is. Different lines are at different angles, and slope is a way of talking about these different slants. One of the most common definitions of slope is "rise over run." What does this mean?
Suppose we want to find the slope between two points. The run is the horizontal separation from the left to the right. This is always positive from left to right. The rise is the vertical separation between the two points as we go from left to right. If the point on the right is higher, than the rise is positive; if lower, it's negative.
Suppose we want to find the slope between (2, 2) and (6, 5). We draw or imagine a little "slope triangle." So this slope triangle allows us to see. There's a horizontal separation and there's a vertical separation. The horizontal separation, the run is four, and the vertical separation is three.
And we could see this just from the legs of the slope triangle. So the rise of three over the run of four, rise over run. So this line has a slope of 3/4, 3 over 4. Suppose we wanna find the slope between (-4,2) and (5,-1). So we draw or imagine this slope triangle, and notice that the vertical leg is 3.
And in fact, since it's higher on the left, it's gonna be a drop. So we have a quote on quote rise of -3, and then the run is that long horizontal leg that has the length of 9. So the slope rise over run, -3/9. And of course, we can simplify this fraction to -1/3, and that is the slope, a slope of -1/3.
If you are given the numerical coordinates of two points, I would suggest using the slope triangle to figure out the slope, in other words, thinking about it visually with the slope triangle. Sometimes, you are given algebraic information, and you have to use a formula for the slope, but do not make the formula your default. This is one of many examples in which over-reliance on the formula produces shallow mathematical understanding.
You actually understand it much more deeply when you're thinking about it visually. After all, the xy plane is visual, it's something you see. Having warned you about the perils of over-reliance on the formula, I will give you the formula. Suppose we have two general points and let's say the one on the left is x1, y1, the one on the right is x2,y2.
It actually doesn't really matter which one's on the right or which one's on the left. The run is gonna be x2- x1, the rise is gonna be y2- y1, and the slope is gonna be rise over run. So we get this formula, y2- y1/x2- x1. So yes, that formula is handy in certain situations, but if you just think about it in terms of rise and run, you'll have a much deeper understanding.
So far, we have been talking about the slopes of two individual points. A line has a slope, and the slope of a line is the slope between any two points on the line. So that's a big idea, we could pick any two points, any two points at all of the whole infinity of points on that line. And we'd always find the same slope between those two points.
If a line has a slope of m = 1/2, then we will pick the two points on the line. And the ratio of rise over run will always simplify to 1/2. So in fact, if you think about the different slope triangles that you would draw on this line, they would all be similar triangles, so they would all have the same ratio. Let's think about a slope of m = 2.
This could mean that we are moving to the right 1 unit, and up 2 units. It could also mean that we move to the right k units and then we have to move up 2k units. It could mean that we move to the left 1 unit and down 2 units. Similarly, we could move to the left k units, and down 2k units. So it might just be a step of 1 and then a step of 2, 1 over and 2 up, but it could be any larger amount in that ratio we could go 5 over and 10 up, or 15 over and 30 up, or something like that.
So for example,if (-3,-1) is a point on a line with a slope of m = 2, then we can find other points. For example if we went over 1 and up 2, so over 1 from -3 would be -2, up 2 from -1 will be positive 1, so that would be the point -2, 1. Or we could start at -3, -1 and we could go to the left 1 and down 2.
So we start at -3 and we go to the left 1, we get to -4, and we start at -1, we go down 2, we get -3. So -4, -3 is also a point on this line. So given a point and a slope, you should be able to find other points. Let's think about a slope of -2/3, a fractional slope. This certainly could mean a run of 3 and a rise of -2, or in other words, to the right 3 units and then down 2 units.
That's one thing it could mean. It could mean to the right 3k units, and down 2k units. Again, as long as we're going in the same proportion, we could go to the right, say 30 units and down 20 units, something like that. Or we might think in terms of fractions, we might move over 1 unit and then go down 2/3 of a unit.
And then we'd think about fractional movements downward. We could also go to the left, we could go to the left 3 units and up 2 units, or of the left 3k units and up 2k units, or of the left 1 unit and up 2/3 of a unit, so there's lots of different things that a slope of -2/3 can mean. Here's a practice problem, pause the video and then we'll talk about this.
Okay, if a line goes through the (2, -1) and has a slope of m = 5/3, find all the points (a, b) on the line where a and b are integers whose absolute values are less than or equal to 10. So we're gonna start at that point. 2,1, and we're gonna move to the right 3, and up 5.
Now, notice if we moved any fraction over, then we've learned our point in the line, but it won't be both the x and the y coordinates integers. So we have to move 3 to the right and 5 up. And so we start at -2, 1. 3 to the right of -2 would put us at 5. And 5 up from -1 will put us at 4, that's 1 point, then we can move 2 to the right and 5 up again, add 3 to the x-coordinate, add 5 to the y-coordinate.
That brings us to the point x (8, 9). If we do it again, then we'd get points whose absolute values are greater than 10, so that doesn't count. We found two points that work and then we ran it to a point that doesn't work. Well, we can also move to the left, that is left 3 and down 5. So we start at 2, -1.
Remove or subtract 3 from the x-coordinate and move down to -1, subtract 5 from the y-coordinate, move down to -6, so -1, -6, that's the point of the line. If we do this again, we're gonna go down to y = -11, again, -11 has an absolute value greater than 10, so this doesn't count. So we found 3 points that satisfy this condition, that are on the line in addition to the point given in the prompt.
Notice that if a line has a slope of 1, then rise equals run, and the slope triangle is a 45-45-90 triangle. That's a really big idea. You should know that lines with slopes of 1 or -1 make 45 degree angles with the each one of the axes. That's really important.
You don't have to worry about the exact angles formed by any other slanted lines. Simply notice that if the slope is greater than positive 1 or less than -1, then you're gonna have an angle greater than 45 degrees. And of course, that's gonna be steeper than any road that you typically would walk on or drive on. What is true of the slopes if two lines are parallel?
If the lines are parallel, they must rise in sync with the same rise for the same run. In other words, parallel lines have equal slopes. That's a big idea, parallel lines always have equal slopes. What is true if the slopes of two lines are perpendicular? First of all, if one line goes up, then the other must go down.
In other words, the slopes must have opposite signs. If one is positive, the other must be negative. The opposite signs is part of the answer, but we need to think about the numerical value of the slope. Think about what happens when we rotate the slope triangle by 90 degrees. So there we have an original slope triangle and then it's rotated by 90 degrees.
So what was the original rise becomes the run. And what was originally the run becomes the rise. So the rise and the run switch places. In other words, the numerator and the denominator of the slope fraction have switched places. Well, that's a reciprocal.
That's what we call a reciprocal when you switch the place of the numerator and the denominator. The slopes of perpendicular lines are opposite-signed reciprocals. In other words, if the slope of the original line is p over q, then the slope of the perpendicular line, make the positive negative, and we flip the fraction over.
If the original slope is 1/2, the perpendicular slope is -2. Slope is rise over run. We find the slope between two points with a slope triangle, or with the slope formula. And again, I would urge you to think more in terms of the slope triangle and not rely too heavily on the slope formula.
The slope of a line is the slope between any two points on the line. Lines with slopes of positive or negative 1 make 45 degree angles with the axes. Parallel lines have equal slopes. And perpendicular lines have slopes that are opposite signed reciprocals of one another.
Read full transcriptSuppose we want to find the slope between two points. The run is the horizontal separation from the left to the right. This is always positive from left to right. The rise is the vertical separation between the two points as we go from left to right. If the point on the right is higher, than the rise is positive; if lower, it's negative.
Suppose we want to find the slope between (2, 2) and (6, 5). We draw or imagine a little "slope triangle." So this slope triangle allows us to see. There's a horizontal separation and there's a vertical separation. The horizontal separation, the run is four, and the vertical separation is three.
And we could see this just from the legs of the slope triangle. So the rise of three over the run of four, rise over run. So this line has a slope of 3/4, 3 over 4. Suppose we wanna find the slope between (-4,2) and (5,-1). So we draw or imagine this slope triangle, and notice that the vertical leg is 3.
And in fact, since it's higher on the left, it's gonna be a drop. So we have a quote on quote rise of -3, and then the run is that long horizontal leg that has the length of 9. So the slope rise over run, -3/9. And of course, we can simplify this fraction to -1/3, and that is the slope, a slope of -1/3.
If you are given the numerical coordinates of two points, I would suggest using the slope triangle to figure out the slope, in other words, thinking about it visually with the slope triangle. Sometimes, you are given algebraic information, and you have to use a formula for the slope, but do not make the formula your default. This is one of many examples in which over-reliance on the formula produces shallow mathematical understanding.
You actually understand it much more deeply when you're thinking about it visually. After all, the xy plane is visual, it's something you see. Having warned you about the perils of over-reliance on the formula, I will give you the formula. Suppose we have two general points and let's say the one on the left is x1, y1, the one on the right is x2,y2.
It actually doesn't really matter which one's on the right or which one's on the left. The run is gonna be x2- x1, the rise is gonna be y2- y1, and the slope is gonna be rise over run. So we get this formula, y2- y1/x2- x1. So yes, that formula is handy in certain situations, but if you just think about it in terms of rise and run, you'll have a much deeper understanding.
So far, we have been talking about the slopes of two individual points. A line has a slope, and the slope of a line is the slope between any two points on the line. So that's a big idea, we could pick any two points, any two points at all of the whole infinity of points on that line. And we'd always find the same slope between those two points.
If a line has a slope of m = 1/2, then we will pick the two points on the line. And the ratio of rise over run will always simplify to 1/2. So in fact, if you think about the different slope triangles that you would draw on this line, they would all be similar triangles, so they would all have the same ratio. Let's think about a slope of m = 2.
This could mean that we are moving to the right 1 unit, and up 2 units. It could also mean that we move to the right k units and then we have to move up 2k units. It could mean that we move to the left 1 unit and down 2 units. Similarly, we could move to the left k units, and down 2k units. So it might just be a step of 1 and then a step of 2, 1 over and 2 up, but it could be any larger amount in that ratio we could go 5 over and 10 up, or 15 over and 30 up, or something like that.
So for example,if (-3,-1) is a point on a line with a slope of m = 2, then we can find other points. For example if we went over 1 and up 2, so over 1 from -3 would be -2, up 2 from -1 will be positive 1, so that would be the point -2, 1. Or we could start at -3, -1 and we could go to the left 1 and down 2.
So we start at -3 and we go to the left 1, we get to -4, and we start at -1, we go down 2, we get -3. So -4, -3 is also a point on this line. So given a point and a slope, you should be able to find other points. Let's think about a slope of -2/3, a fractional slope. This certainly could mean a run of 3 and a rise of -2, or in other words, to the right 3 units and then down 2 units.
That's one thing it could mean. It could mean to the right 3k units, and down 2k units. Again, as long as we're going in the same proportion, we could go to the right, say 30 units and down 20 units, something like that. Or we might think in terms of fractions, we might move over 1 unit and then go down 2/3 of a unit.
And then we'd think about fractional movements downward. We could also go to the left, we could go to the left 3 units and up 2 units, or of the left 3k units and up 2k units, or of the left 1 unit and up 2/3 of a unit, so there's lots of different things that a slope of -2/3 can mean. Here's a practice problem, pause the video and then we'll talk about this.
Okay, if a line goes through the (2, -1) and has a slope of m = 5/3, find all the points (a, b) on the line where a and b are integers whose absolute values are less than or equal to 10. So we're gonna start at that point. 2,1, and we're gonna move to the right 3, and up 5.
Now, notice if we moved any fraction over, then we've learned our point in the line, but it won't be both the x and the y coordinates integers. So we have to move 3 to the right and 5 up. And so we start at -2, 1. 3 to the right of -2 would put us at 5. And 5 up from -1 will put us at 4, that's 1 point, then we can move 2 to the right and 5 up again, add 3 to the x-coordinate, add 5 to the y-coordinate.
That brings us to the point x (8, 9). If we do it again, then we'd get points whose absolute values are greater than 10, so that doesn't count. We found two points that work and then we ran it to a point that doesn't work. Well, we can also move to the left, that is left 3 and down 5. So we start at 2, -1.
Remove or subtract 3 from the x-coordinate and move down to -1, subtract 5 from the y-coordinate, move down to -6, so -1, -6, that's the point of the line. If we do this again, we're gonna go down to y = -11, again, -11 has an absolute value greater than 10, so this doesn't count. So we found 3 points that satisfy this condition, that are on the line in addition to the point given in the prompt.
Notice that if a line has a slope of 1, then rise equals run, and the slope triangle is a 45-45-90 triangle. That's a really big idea. You should know that lines with slopes of 1 or -1 make 45 degree angles with the each one of the axes. That's really important.
You don't have to worry about the exact angles formed by any other slanted lines. Simply notice that if the slope is greater than positive 1 or less than -1, then you're gonna have an angle greater than 45 degrees. And of course, that's gonna be steeper than any road that you typically would walk on or drive on. What is true of the slopes if two lines are parallel?
If the lines are parallel, they must rise in sync with the same rise for the same run. In other words, parallel lines have equal slopes. That's a big idea, parallel lines always have equal slopes. What is true if the slopes of two lines are perpendicular? First of all, if one line goes up, then the other must go down.
In other words, the slopes must have opposite signs. If one is positive, the other must be negative. The opposite signs is part of the answer, but we need to think about the numerical value of the slope. Think about what happens when we rotate the slope triangle by 90 degrees. So there we have an original slope triangle and then it's rotated by 90 degrees.
So what was the original rise becomes the run. And what was originally the run becomes the rise. So the rise and the run switch places. In other words, the numerator and the denominator of the slope fraction have switched places. Well, that's a reciprocal.
That's what we call a reciprocal when you switch the place of the numerator and the denominator. The slopes of perpendicular lines are opposite-signed reciprocals. In other words, if the slope of the original line is p over q, then the slope of the perpendicular line, make the positive negative, and we flip the fraction over.
If the original slope is 1/2, the perpendicular slope is -2. Slope is rise over run. We find the slope between two points with a slope triangle, or with the slope formula. And again, I would urge you to think more in terms of the slope triangle and not rely too heavily on the slope formula.
The slope of a line is the slope between any two points on the line. Lines with slopes of positive or negative 1 make 45 degree angles with the axes. Parallel lines have equal slopes. And perpendicular lines have slopes that are opposite signed reciprocals of one another.
