Today I’m going to show you yet another reason why I keep telling you as soon as you see a linear equation to begin by solving for Y. Remember, the rectangular coordinate system began with the x-axis had 0, the positive numbers, and the negative numbers to the left. That’s the x-axis. The y-axis of course went vertically with the positive numbers going up and the negative numbers going down. OK? Now, let’s look at this line, the line represented representing the equation Y=2X-1 That’s it. You just have to trust me. That point right there is where it’s intersecting the y-axis, isn’t it? Well that point is called its y-intercept. That is just what it is. It’s where it intersects the y-axis Let’s look at another. This equation: y equals -3x plus 4. Let’s solve for y. If I graphed it, it would be right here. Now where is it intersecting the y-axis? Its y-intercept is 4 because that’s where it intersects the y-axis. You might want to note that its X intersept, intercept is one-and-a-half but for right now we’re only talking about the y-intercept. Now let’s go back to this line. We know the y-intercept is minus 1. I want you to note something. What is the value of x at the y intercept? In other words where are you on the X line? Well, it’s right in the middle, isn’t it? It’s 0 and if you put 0 into this equation for x. Of course, the part of it will cancel out and you’ll get the constant y equals minus 1 well that was the y-intercept, wasn’t it? The point 0-1 We’re going to have a pattern here. We go back to the second equation and put in 0 for X, what are we going to be left with? The constant so you could say if you’d solve for y that the number, the constant sitting by itself in that equation will always be the y intercept, the other number, the number in front of x will always be the slope. You don’t have to go to any trouble if you’ve solved for y. You don’t have to go to any trouble of subtracting and dividing and all that if you’ll solve for y that number is always the slope. So in summary if we solve an equation for y, it is in what’s called slope-intercept form. It makes sense doesn’t it? Where the number in front of X is always the slope and the other number is always the y-intercept. For instance, in this equation, now that I’ve solved for y the slope is seven and the y-intercept is three. Just that easy! In this equation, the y-intercept is seven and the slope is minus 2 and note I said the number in front of X is the slope not the first number, the number in front of x. So if I switch it negative 2 is still the y-intercept and six is still the slope, the number in front of X. How about this one? Well it’ll still work. Seven is the slope. What’s the y-intercept? Well, what’s the number sitting by itself, the constant? You could have written plus 0. OK. So you kind of have to imagine that and the y-intercept will be 0. In this one you kinda have to imagine a number in front of X. We know the number sitting by itself is the y-intercept. What would the slope be, the number in front of x? You could write it like this, couldn’t you? The slope is 0 and if you think about it, y equals 3, the graph does have a slope of 0. Y equals 3; it’s a flat line isn’t it? How about the slope and y-intercept of this equation? Remember you can’t do anything until you solve for y. Now who’s keeping y from being by himself? The 6x so I’ll subtract 6x from both sides which I really can’t do on the right. I just write eight minus 6x. They’ll be telling me 2 something something and on the left side I still have to get rid of that 2 that’s multiplying the Y. So I’ll divide both sides, everybody on both sides and I’ll get y equals 4 minus 3x. Now it’s easy. The slope is the number in front of X minus 3 and the y-intercept is the other number 4. Got it? Let’s work backwards. I want you to give me the equation this time. Give me the equation of a line with slope 5 and y-intercept minus 3. Remember it’s always going to be of this form where the slope is the number in front of X and the y intercept is the number sitting by itself, the constant. So there’s your answer: y equals 5x minus three. Very easy if you know the form. Even if I said give me the equation of slope dadadada and Y heeee and I need to know this. the equation will be y equals wococo x plus bluuuuha. Just fill them in. Get the idea? How about finding the equation of a line with slope 2 that contains the point (1,5)? Well right away I’d take the 2 and put them into the slope but I still have to find the b but I do have added information here. I have a point that fits. What I can do is make it fit. Remember the first number is the x and the second number is the y. So let’s put the 1 in for the x and the 5 in for the y and make it fit. Well, if five equals 2 times 1 plus b or 2 plus B, what is does B have to be? Subtracting 2 from both sides b will have to be three. And that’s not the answer. The answer is the equation. So I take what I know to be the slope which is 2 and what I know to be the B or y-intercept which is 3 and I get the equation. Now wait a minute, why would we even want the equation? Because once you have the equation you can find other points of this function if you would. You can find all kinds of points once you have that general equation and that’s what we’ll use it for. This may be the equation for converting amongst temperatures or the equation for predicting stock prices or once you have that equation you can use it. Let’s try it again. Make sure you can follow. To get the equation of a line with the following clues: a slope of negative 3 and it contains the point negative (-2,8). It’s got negatives coming out all over the place. When I use my general equation, put in the slope that they give me and now I’ve got to make that b fit. Let’s put in the first coordinate for X and the second coordinate for Y Let’s see, negative 3 times negative two is six so I’ll solve for b. Probably do it in your head. And b is going to have to be 2 but remember they asked for the equation. So there’s your answer. We could use this to graph lines. When I graph the line containing the y-intercept of two and a slope of minus 1 over 2. Well that has a y-intercept of two. That’s the equation and that’s the y-intercept because it hits the y-axis at 2 and remember what the slope means. Between any two points I’m going to execute the slope of going down 1 and over 2 and find another point. So really if I had the equation I could draw the line just that quick. How cool is that! That’s going to be very useful. Let’s try it again. Let’s draw the graph of the line that has a y-intercept of 1 and a slope of 3. Right away we can come up with the equation: y intercept of 1 and a slope of 3. The three is in front of the x and the 1 is the constant. Well, I can put the y-intercept. What does it mean to be the y-intercept? It means it hits the y-axis at 1 so the point (0,1) is on this line. The other thing is that the slope is 3 and 3 as a fraction is three over one, isn’t it? So up three and over 1 and we’ll have another point. Once we have two points you’re there and you could go up three and over one from the next point also but you’d still be on that line. So, you’ve got the line and the equation to boot. Let’s try and do two at a time. If you consider drawing two lines at a time, three things can happen. One is that they could look like this and be what we call parallel, never meet or they could intersect. Right? These two lines intersect. What’s the other third possibility? If you don’t guess it is that the two lines are actually the same line. That could happen too but doesn’t happen very often. So those are the three possibilities: parallel means no points in common, intersect means one point in common and same line of course as you can see have infinite number of points in common. I’m gonna have to ask you given the equations which of the three is going on here? And you should be able to tell me actually without drawing the picture. Well, we know we can’t do anything until we solve for y. I’ll solve the first one for y subtracting 6x. And we’ve done this one before and divide everybody by two and there’s that one solved for y. And this one divide everybody by four and we have y equals four minus three x versus y equals 2 minus 3x. Well, we’re not even going draw the picture, I know that the slopes are the same. Think about two lines that are leaning the same way. They have the same slope but they’re different. Now they’re different lines because the y-intercepts are different. Two lines that have the same slope but different y-intercepts. I didn’t even have to draw them. I know they’re parallel. That’s what it means to be the same slope. Let’s try some more. This line and this line. We can’t do anything until we solve for y. So let’s subtract 6x + divide by 2. You get pretty good at this after a while. Let’s take a look at this one, I’m going to subtract 12x and I guess I’ll have to divide everybody by four and this might stare you’re right in the face. Slopes are the same and the y-intercepts are the same. You might not have guessed it but those lines really are the same line. You couldn’t tell anything until you solve for y, could you. Let’s do one more. Let’s solve the first one for y. You can’t do anything until you solve for y. Who’s keeping y from being alone here? It’s the two. That’s all; he’s the only troublemaker, Dividing by 2 I get y equals 3x plus 4 subtracting here and dividing I get y equals 2 minus 5x over 2. Now that looks difficult but if you think about it all that matters is that the slopes are different. As soon as the slopes are different, 2 lines that have different slopes considering the lines go on forever they have to touch. They may touch in heaven or hell but they have to touch and if they touch they intersect. They intersect in one point so the point of this is you can do this; you can analyze these problems without even drawing a picture. So you need to practice that. Let’s go do the homework.