Geometry Slope of Lines

Hello--welcome back to Educator.com.0000

The next lesson is on the slope of lines; this might be a little bit of a review for you from algebra.0002

But this whole lesson is going to be on slope.0010

Slope is the ratio between the vertical and the horizontal, or we can say "rise over run."0014

Rise, we know, is going up and down; and then, the run is going left and right.0027

So, when we talk about slope, we are talking about the vertical change and the horizontal change.0033

For slope, we use m; so the slope of a line containing two points with coordinates (x₁,y₁)0044

and (x₂,y₂) is given by this formula right here.0057

Now, (x₁,y₁), these numbers right here, and (x₂,y₂), those numbers, are very different from exponents.0061

They are not to the power of anything; it is just saying that it is the first x and the first y.0073

So, we know that all points are (x,y), and so, this right here is just saying that this is the first x and the first y.0083

And this is also x and y, but you are just saying that it is the second point; it is the second x and the second y.0094

Because you have two x's and two y's, you are just differentiating the points; this is the first (x,y) point, and this is the second (x,y) point.0103

It doesn't matter which one you label as first and which one you label as second.0112

You are just talking about two different points.0116

And when you have two points, then the slope is going to be (y₂ - y₁)/(x₂ - x₁).0118

You are going to subtract the y's, and that is going to be your vertical change, because y, you know, is going up and down.0130

And x₂ - x₁ is your horizontal change, the difference of the x's, which is going horizontally.0139

Now, it doesn't matter...like I said, if you are going to subtract this point, this second y, from this y,0150

then you have to make sure that you subtract your x's in the same order.0163

If you are going to subtract y₂ - y₁, then it has to be x₂ - x₁ for the denominator.0168

It can't be (y₂ - y₁)/(x₁ - x₂).0174

If you do y₂ - y₁ over here, you cannot do x₁ - x₂.0179

You can't switch; it has to be subtracted in the same order, or else you are going to get the wrong answer.0185

And this right here is just saying that x₁ and x₂, these numbers, can't equal each other,0190

because if they do, then this denominator is going to become 0.0197

If x₁ is 5, and x₂ is 5, then it is just going to be 5 - 5, and that is going to be 0.0201

And when we have a fraction, you can't have 0 in the denominator, or else it is going to be undefined.0212

So, that is what it is saying right here: they should not equal each other, or else you are going to have an undefined slope.0218

Let's find the slope of these lines: here we have (-4,-2) and (5,3).0229

Now, you can do this two ways: you can use the slope formula by doing (y₂ - y₁)/(x₂ - x₁);0239

if you have a coordinate plane, and it is marked out for you--you have grids that show each unit--0252

then you can count; you can just go from one point to the other point,0262

and you can just count your vertical change and count your horizontal change; you could do it that way.0269

But since these are not labeled--each unit is not labeled out--let's just use the slope formula.0275

Here, if I make this (x₁,y₁), (x₂,y₂), then my slope is going to be (-2 - 3)...0285

so then, this value is (y₂ - y₁, which is 3), over (-4 - 5).0301

Now, I could do (3 - -2); I can go that way if I want, but if I do that, if I choose to do this one first,0312

(3 - -2), then I have to do (5 - -4); you have to be in the same order.0321

If you do 3 minus -2, then you can't go with (-4 - 5); you can't go the other way then.0327

It doesn't matter which one you start with; but when you do your x, you have to do it in the same order.0338

This one right here is -5/-9, which is just 5/9.0345

Now, without solving slope, if you look at the line, you should be able to tell if the slope is going to be positive, negative, 0, or undefined.0358

For this one, since the slope measures how slanted a line is, how tilted a line is, if we look at this line,0374

imagine a stick man (I like to call him "stick man," because I can only draw stick figures) walking on this line.0388

Now, he can only walk from left to right, because let's say you read this--you have to read from left to right.0398

So then, it can only go from left to right; he is walking uphill, and that would be a positive slope.0407

This is a positive slope; if the stick man is walking uphill, it is a positive slope.0416

If the stick man is walking downhill, like the next one (again, he can only walk from left to right)--he is going to walk downhill, so this is a negative slope.0426

Without even solving, I know that my slope is going to be negative.0439

This is positive; it is positive 5/9.0443

Now, the slope for this one--I know, before I even solve it, that it is going to be negative.0446

So, after I do solve it, if I get a positive answer, then I know that I did something wrong, because it has to be negative.0451

For this one, the slope is 5 - -4; and make sure that you are going to find the difference of the y's for your numerator.0463

Don't do your x's first; the numerator is y's; the denominator is x's.0475

I went from this to this, so then I have to do -2 - 3.0482

So then, this is...minus a negative is the same thing as a plus, so 5 + 4 is 9, and then -2 - 3 is -5.0489

So, this is -9/5; and I have a negative slope, so that is my answer.0502

A couple more: here I have a horizontal line; my slope is (y₂ - y₁)...(-3 - -3), over (-6 - 4); this is 0,0519

because -3 + 3 is 0; this is -10; well, 0 over anything is always 0; so the slope here is 0.0541

Now, let's bring back the stick man: if stick man is walking on this, he is not walking uphill or downhill; he is just walking on a flat surface.0554

If he is walking on a flat surface, since slope measures how slanted a line is, it is not slanted at all--it is just flat; that is why the slope is 0.0563

Whenever it is flat, it is a horizontal line, and the slope will be 0--always.0573

It doesn't matter if it is up here or down here; as long as it is a horizontal line, your slope is going to be 0.0580

The next one: 4 - -4...be careful with the negatives: it is 4 minus -4; -2 - -2; change those to a plus--0592

minus a negative is also a plus--so 4 + 4 is 8, over -2 + 2...is 0.0615

Now, look at the difference between this one and this one.0624

In this one, the 0 is in the numerator; if it is in the numerator, it makes it just 0; 0 is a number, just like 5, 6, -3;0628

those are all numbers, and 0 is a number; so your answer for this problem, your slope, is 0; you have a slope; it is 0.0640

And in this problem, we cannot have 0 in the denominator--it is just not possible.0650

So, since you did come up with a 0 in the denominator, this is going to be undefined.0657

You can also say "no slope"; in this case, you do have a slope--the slope is 0; in this case, you do not have a slope.0669

There is no slope; it doesn't exist; it is undefined, because the denominator is 0.0680

So, my answer is just "undefined."0685

And then, to bring back the stick man: since it is a man, it can't do this--this is like walking up a wall.0689

Stick man can't walk up a wall; it is not possible--it can't do it.0702

He can't walk up a wall; he is not Spiderman; he can't walk up a wall.0706

So, in this case, this man can't do this; if he can't walk up this wall, it is undefined; it can't be done; there is no slope.0711

If he is walking on a horizontal--no slant at all--it is 0.0724

If he has to walk up a wall (which is impossible), then it is an undefined slope.0731

If the stick man is walking uphill, it is a positive slope; downhill is a negative slope; a horizontal line is 0; a vertical line, like a wall, is undefined.0737

You can't walk up a wall.0749

A couple of postulates: If we have two non-vertical lines that have the same slope, then those lines are parallel,0754

because again, slope measures how slanted a line is.0769

So, if I have two lines that are slanted exactly the same way, then they are going to be parallel.0775

Again, two lines that have the same slope are parallel.0788

And this part right here: "if and only if"--now, we went over conditionals, if/then statements;0794

to change this one right here (let's go over this...number 1)...two non-vertical lines have the same slope if and only if they are parallel.0807

This just means that this conditional and its converse are both true.0819

It is basically two statements, two conditionals in one.0833

I can say, "If two lines have the same slope, then they are parallel."0840

And this would be the converse: I can say, "If two lines are parallel, then they have the same slope."0865

The statement and its converse are both true: this is true, and this is true.0893

So, just instead of writing each of those conditionals separately, you can write them together by "if and only if."0897

It just means that this statement and its converse (converse means, remember, that you switch the hypothesis and the conclusion) are both true.0908

So then, you can just use "if and only if."0919

If two lines have the same slope, then the two lines are parallel.0925

Or you can say, "If two lines are parallel, then they have the same slope."0930

Either way, parallel means same slope; same slope means parallel.0934

Now, the next postulate is talking about perpendicular lines: Two non-vertical lines are perpendicular if and only if the product of their slopes is -1.0939

Now, "the product of their slopes is -1"--that means that, if, first of all, I have a line like this,0957

and I have a line like this, let's say they are perpendicular; and the slope of this line, let's say, is 1/20970

(it has to be positive; it is going uphill); then the slope of this line is going to be the negative reciprocal,0979

meaning that you are going to make it negative; if it is negative already, then you are going to make it positive;0990

so, the slope of this line will be negative...and then the reciprocal of it will be 2/1.0997

So then, it is saying that the product of their slopes is going to be -1; so 1/2 times -2 is -1.1006

Just think of it as: If you have two perpendicular lines, then the slopes are going to be negative reciprocals of each other.1023

And if you multiply those two slopes, then you should get -1, always.1032

OK, parallel or perpendicular lines: you are given points A, B, C, and D; you want to determine if line AB is parallel or perpendicular to CD.1040

So, to determine if the two lines are parallel or perpendicular, then you have to compare their slopes.1058

So, for line AB, I need to use points A and B.1066

If I find the slope using these two points, the slope of AB is going to be -6 - 0, y₂ - y₁, over x₂ - x₁.1070

And that is -6/-3, so we have 2.1093

And then, the slope of CD is y₂, -3, minus -4, over 4 - 2; this is 1/2, so this is positive 1 over 2.1100

Did I get that right?--yes.1133

In this case, it is going to be neither, because here we have AB; (-6 - 0)/(-2 - 1) becomes positive 2.1138

And then here, this is -3 - -4, and 4 - 2; for this, I get positive 1/2.1157

Now, it looks like they are going to be perpendicular, but remember: they have to be the negative reciprocal of each other.1168

If this is 2, this is 2/1, and the inverse, or the reciprocal, is 1/2; but they are not negative--it is not negated.1176

So, if I multiply 2/1 times 1/2, I am only going to get 1, not -1; so this is neither.1188

OK, find the value of x if the line that passes through this point and this point is perpendicular to the line that passes through (-1,6) and (-2,8).1204

We are given our two points that we have to find the slope of.1220

But from those two points, one of the values, the x-value, is missing.1227

That means that we need the slope.1235

They didn't just give us the slope in this problem; they didn't just hand it to us.1239

We have to actually solve for the slope, because we know that it is perpendicular to a line that passes through these two points.1243

So, basically, the points that I have to use are (x,4) and (-3,3).1255

I have to find x; and this is another line, and I am just going to use that line to find the slope,1266

because I have my slope that I need that has a relationship with the slope of this line.1278

So, to find the slope of this line right here that passes through these points,1285

I am going to do (6 - 8), (y₂ - y₁), over (-1 - -2); this is -2/1, so the slope is -2.1293

But since I know that my line is perpendicular to this line, my slope is going to be the negative reciprocal of this slope.1307

If this slope is -2, then my slope is going to be positive 1/2, positive one-half.1322

That is what I need to use: the slope is positive 1/2.1333

Now, using the slope formula, I know that this is (y₂ - y₁)/(x₂ - x₁).1338

Well, I can just fill everything in, except for this, and then use that as x₁.1350

1/2 is my m; y₂...if this is (x₁,y₁), (x₂,y₂),1356

y₂ is 3, minus 4, over -3, minus x; since I don't know this value, which is what I am solving for, I can just leave it like that.1365

And then, from here, I have to solve this out.1384

I can solve this out a couple of ways: first of all, since this is a fraction equaling a fraction, I can use proportions.1388

I can make (-3 - x) times 1 equal to 2 times 3 minus 4; or let me just do this--let me just simplify this first.1399

1/2 = -1/(-3 - x); or I can just multiply...1411

I have a variable; the variable that I am solving for is x, and that is in the denominator.1423

If I want to solve for the variable, it cannot be in the denominator, so I have to move it out of the denominator.1427

I can do that by multiplying both sides or the whole thing by -3 - x;1433

or again, since this is like a fraction equaling a fraction, like a proportion, I can just do that.1439

So, just make -3 - x equal to...and then, I am just multiplying it this way...equaling this; it is -3 - x = -2;1444

if I add 3, then I get -x = 1; x = -1.1459

So again, you are going to find your slope.1469

They might not just hand you the slope; they might not tell you what the slope is directly.1476

So then, you have to find it this way; they will give you another line that has a relationship with your line, your slope.1483

So, you have to find the slope of that other line, and then use that slope to find your slope.1496

And then, plug it all into the slope formula; and then from there, you just solve.1503

Let's do a few examples: Find the slope of the line passing through the points.1510

Again, here is the slope formula; this equals...it doesn't matter which one I use first, so I will just use (-2 - 5) first.1517

That means that I have to use this one first; so it is (3 - -4).1536

This becomes -7/7, which is equal to -1; and then, for this one, the slope is (0 - 6)/(-7 - -7).1542

This is going to be -6/0; 0 is in the denominator, which means that I have an undefined slope.1564

And that just means that the line that is passing through these points is going to be a vertical line; vertical lines have undefined slopes.1580

Find the slope of each line: now, they don't give you the points--they just show me the lines.1593

And I have to see what points the lines are going through to find the slope.1602

Let's see, let's look for the slope of n first; here is n.1612

Now, remember: for slope, I can do this two ways: I can find two points on this line, like this and like this--1618

those are two points on the line (or here is another point; it doesn't matter--any two points on the line);1630

you can find the coordinates of the points and use the slope formula.1636

For two points, find the coordinates and use the slope formula.1642

Or an easier way, in this problem: Since we have all of these grids marked out for us, I can just1646

(because slope is rise over run, the vertical change over the horizontal change; rise is how many it is going up or down,1656

and then run is how many is going left or right)--whenever I go up (here, this is the positive y-axis), any time I am counting upwards,1668

it is a positive number; if I am counting downwards, then it is a negative, because I am going smaller.1683

If you are going up, it is a positive number; if you are going down, if you are counting down, then it is a negative number.1690

The same thing for x: if you are moving to the right, it is a positive number; if you are moving to the left, it is a negative number,1696

because it is getting larger as you go to the right; and as you go to the left, you are moving towards the negative numbers.1703

You are getting smaller, so it is negative if you are going to the left.1709

To find the slope of n, I am just going to do rise over run; I am going to just count my vertical and horizontal change.1715

You go from any point to any other point on the line.1724

I can start from here; I am going to go one up, because it is on this right here.1729

My vertical change: I only went up one; remember, up is positive, so to find the slope of n, it is positive 11740

(that is my rise), and then I am going 1, 2 to the right--that is positive 2, so the slope is 1/2.1754

Now, remember: I can also go from any point to any other point on the line.1767

So, if I start from this point, let's say I am going to go from this point, and then (I didn't see this point, so) to this point;1773

then I can go down 2 (remember: down 2 is negative 2), over...then I am going to go 1, 2, 3, 4.1781

And that is to the left, so that is negative 4; -2/-4 is the same thing as 1/2.1797

It doesn't matter how you go from whichever point to any other point on the line, as long as both points are on the line,1808

and as long as you make it a positive number going up, a negative number going down, positive to go right, and negative to go left.1817

You are going to get the same answer; you are going to get the same slope.1826

The slope of n is 1/2; then the slope of p (let me use red for this one): let's see, I have a point here, and I have another point here.1830

So again, I can go from this point to that point, or I can go from that point to that point; it doesn't matter.1853

Let's start right here: I am going to go 1, 2, 3; I went up 3, so that is a positive 3.1858

And then, from here, I go to the left 1, which is a negative 1.1869

3/-1 is the same thing as -3, so my slope of p is -3.1876

Or I could go from this point to this point; that would be to the right one (that is positive 1), over down 3 (1, 2, 3);1883

oh, I'm sorry; I did horizontal over vertical, which is wrong; so I have to go this way--vertical first.1893

1, 2, 3: that is a -3, and to the right 1--that is positive 1; so this is also -3, the same thing.1902

The next line is line q (I will use red for this one, too): this is a vertical line.1916

Automatically, I know that that has an undefined slope. I can also just...1932

Now, I know that this line is not really completely lined up with the grid, but sometimes when you transfer1939

this into this program, or move things into this program, it might shift a little bit.1947

But think of this line as being on 2 right here, as x being 2.1951

Let's say I have this point right here, and then any other point--that point right there.1960

All that I am doing is: my vertical change is going down to -2; my horizontal is nothing: 0.1964

I am not moving to the right or left at all; that is 0.1975

So, we have a 0 in the denominator; this is an undefined slope.1979

And again, I knew that because this is a vertical line; the stick man can't walk up that line; so it is an undefined slope.1989

And the last one, for line l: any two points...1999

Again, this line is shifted a little bit, but I can just do that if I want.2007

Vertical change first: the vertical change is 0, because I am not moving vertically; to get from this point to this point, I don't go up or down at all.2014

So, it is 0 over...and then, I can move 1, 2, 3 to the right; so no matter what the bottom number is, my slope will be 0.2021

The slope of the l is 0; again, it is a horizontal line, so it is not slanted; it is not going uphill or downhill--nothing.2035

It is just horizontal; then the slope is 0.2044

The next example: Graph the line that satisfies each description; slope is 2/3 and passes through (-1,0).2052

You just have to graph this first one; let's say I am going to graph it right up here.2066

Now, just a sketch will do; let's say...here is my x; here is my y; (-1,0) is right there.2075

My slope is 2/3; so again, this is rise over run.2103

Now, I can use the same concept, the positive going up and negative going down, positive to the right and negative to the left.2110

The top number, the rise, to go up and down: I have a positive 2--that means I am going to go up 2, because it is positive.2120

From here, I am going to go 1, 2; and then, I have 3 that I am going to move to the left or to the right;2130

but since it is a positive 3, I am going to move to the right: 1, 2, 3.2139

Now, from this point, I can go down if I want to, because 2/3, that slope, is the same thing as -2/-3.2150

So, if I go -2, I am going to go down 2: 1, 2; and then, -3 is to the left, so 1, 2, 3; there is my line, right there.2163

This is the second one: it passes through point (3,1) and is parallel to AB with A at this point and B at that point.2182

Again, they don't give us our slope; they just give us the point that we need to use.2194

Our line is going through this point, and we don't have the slope of our line;2201

instead, they give us the slope of another line, line AB; and they say that it is parallel to it.2205

So, as long as we find the slope of AB, since it is parallel, we know that our slope will just be the same as this slope.2211

The slope of AB is (4 - 3)/(-1 - -2), which is 1 over...this is 1...so 1.2222

Now, since our slope is parallel, again, our slope is 1.2238

And then, this is our point; so we have point (3,1), and the slope is 1.2247

To graph (x,y), (3,1), it is 1, 2, 3; and 1; there is our point that our line is passing through.2258

And then, our slope is 1; 1 is the same thing as 1/1, so positive 1 is up 1, to the right 1; also, positive 1 is up 1.2291

And then again, you can just do -1/-1; that is the same thing as 1.2308

So, from here, I can go down 1, left 1; down 1 is -1; left 1 is -1.2313

And that is going to be a line like that.2322

The last example: Determine the value of x so that a line through the points has the given slope.2331

Again, they give us a slope, and then we have to find the missing value, which is x.2338

Since we know that the slope formula is (y₂ - y₁)/(x₂ - x₁),2345

if I make this (this is (x,y), and this is also (x,y)) my first point, and this is my second point--2354

so that is (x₂,y₂)--then my slope is y₂, which is -5, minus 1, over x₂, -3, minus x.2366

Now, again, you can turn this into a proportion; or I can just multiply this out to both sides.2385

I can multiply this to this right here to cancel it out, and then multiply to the other side and distribute that; I could solve it that way.2397

Or I could just make this a proportion: 2/1 = -6/(-3 - x); so to continue right here, it is going to be 2(-3 - x) = -6.2406

This is -6 - 2x, and you just distribute that; that equals -6; -2x = 0; x = 0.2430

So, we have 0 as our x for this problem.2444

OK, the next one: again, plugging everything into the slope formula, we have 4/3 = (-2 - -6)/(x - 7).2450

So, 4/3 =...this is -2 + 6, so this is 4, over x - 7.2472

OK, well, in this problem, again, you can multiply x - 7 to both sides to get it out of the denominator,2482

because since you are solving for x, it cannot be in the denominator.2492

Or you can cross-multiply using proportions, because it is a fraction equaling a fraction.2497

Or, since this 4 numerator equals this 4 numerator, then the denominator has to equal the denominator, so you can make 3 equal to x - 7.2505

So, let's just solve it out, multiplying: I can multiply this side by (x - 7) like that,2518

and multiply this side by (x - 7); then, it is going to be 4/3x minus 28/3 equaling 4.2529

Now, this is actually probably the harder way to do it; but I just wanted to show you how to multiply it to both sides.2553

This is a binomial that, again, you are just multiplying to both sides.2561

But the easiest way would be to make (x - 7) equal to 3: because 4 = 4 (the top), then 3 = x - 7 there.2567

So, let's just continue out this way: if I add 28/3 to both sides, this is going to be the same thing as 12/3 + 28/3,2577

and that is just because I need a common denominator; that is equal to 4/3x.2593

That equals...this is 40/3...if I multiply the 3's to both sides, this will be 4x = 40; x = 10.2605

So here, x = 10, this value right here.2626

And if we solved it the other way, 3 = x - 7, then you would just add 7, so x would be 10.2632

That is it for this lesson; thank you for watching Educator.com.2642