In this video we’re gonna talk about linear equations. We’ll explain what we mean when we say linear equations. We’ll look at some examples and draw some graphs of linear equations. And then we’ll find links between how the graphs look and the elements of the equation itself.
First let’s think about a real-life situation which is an example of a linear equation. Imagine a taxi that charges you a fixed fee of three dollars for every journey — so that’s basically a price to get into the cab — and then two dollars for every mile that you drive on that journey. Now we can make a linear equation out of this. If we let 𝑥 be the number of miles that we travel and 𝑦 is the total cost of the fare in dollars, then 𝑦 is equal to two times the number of miles that we drive because it’s two dollars per mile plus the three dollars fixed fee that we have for every journey.
Now we can plot a graph of this. So first let’s make up a table of values and then we can plot the points. So we’re just gonna consider cases where the number of miles are between zero and ten. Obviously we could go a lot further, but that’s all we’ll do for now. And we’re gonna look at zero miles, one mile, two miles, three miles, four miles, and so on.
So if we travel no miles, 𝑦 would be two times zero plus the three; that’s equal to three dollars. If we travelled one mile, then 𝑦 would be two times the one, so two dollars for that one mile plus the three dollars to get into the cab; that’s five dollars. If we travelled two miles, it would be two lots of two for the mileage; that’ll be four dollars plus the three for getting into the cab is seven dollars. It will be nine dollars for three miles, eleven dollars for four miles, thirteen dollars for five miles, and so on.
So what you might have spotted is that every time I add one to the number of miles we’re travelling, we’re adding two to the total fare. So add another one to the number of miles we’re travelling here, that adds two to the number or to the amount of money that we have to pay. So to work out for six miles, we just gonna have to add two dollars onto here, which will make it fifteen dollars. And we can fill in the rest of the table much more quickly.
Okay let’s plot that all on a graph and see what it looks like. So I’ve drawn my axes here; my cost of the fare is the 𝑦-axis and the number of miles that we’re travelling is on the 𝑥-axis. So when 𝑥 is zero, 𝑦 is three; that’s this point here, when 𝑥 is one, 𝑦 is five, when 𝑥 is two, 𝑦 is seven, and so on. So I can join those all out with a nice straight line.
And what you can see on a straight line graph is that whether I’m at the beginning of my journey here if I travel an extra mile, it’s going to cost me an extra two dollars. If I’m towards the end of my journey and I travel one extra mile, it’s still going to cost me two extra dollars. So regardless of whether I’m just setting out on a journey or whether I’ve already been twice around the world in this cab, I want to go an extra one mile; that extra one mile is gonna cost me an extra two dollars. And that concept is called the slope of the line or the gradient of the line and is defined as the amount that the 𝑦-coordinate would change if I add one to the 𝑥-coordinate. And the definition of a straight line is that no matter where about you are on that straight line, that figure will always be the same. The slope of the line will not change regardless of what the 𝑥-coordinates are.
So hopefully you noticed that the number that we’re multiplying the 𝑥 by is the slope or the gradient of the line. Now let’s think about this point here, where that line cuts the 𝑦-axis. Now if we think about the coordinates of that point, the 𝑥-coordinate is zero and the 𝑦-coordinate is three. Now if we put this 𝑥-coordinate into our equation, we get 𝑦 is equal to two times zero plus three. Well two times zero is obviously zero. And zero plus three is just three.
Now that value has a special name as well. Whereas straight line cuts the 𝑦-axis or what’s the 𝑦-coordinate when the 𝑥-coordinate is zero, that is called the intercept of the line. So in our particular example, this represents what fare I would pay if I travelled zero miles and that’s three dollars. So provided your equation of your line — wherever that line is — is in this format 𝑦 is equal to something times 𝑥 plus some other number, we can instantly tell what the line is going to look out. We know the slope of the line — in other words what’s happening to the 𝑦-coordinates — as we increase our 𝑥-coordinate by one each time and where does it cut the 𝑦-axis.
So all equations of straight lines look like this: 𝑦 equals some number times 𝑥 plus something else. Now the thing that we’re multiplying 𝑥 by tells us the slope of the line — how steep the line is — and the intercept is the other number, where it cuts the 𝑦-axis. So for example, 𝑦 equals three 𝑥 plus seven; every time I increase my 𝑥-coordinate by one, my 𝑦-coordinate would increase by three. So that would be a steeper line than the one we’re just been looking at. And in this case, it would cut the 𝑦-axis at seven. So relating this back to a taxi example, your fare for just getting in the cab will be seven dollars in this case. And every mile that you travelled would be three dollars.
So let’s think back to our original example and it was a bit restricted because it was based in the real world. For example, we can’t actually travel negative distances. So imagine if you got into a cab and said “please, drive home backwards.” So you might start off here at a distance of zero miles, which would cost you three dollars. And then you wanna drive ten miles home which is negative ten miles because you’re driving backwards. And by the time you’ve driven ten miles, your fare has come down to negative seventeen dollars. That means the cabbie has to pay you seventeen dollars. Well I don’t think that’s a business model that’s gonna last. So in the real world we might be forced to use-use part of a straight line graph, but in theory they go off forever in this 𝑥-direction and forever in this 𝑥-direction.
Right let’s have a look at some more straight line graphs. So this one here, it cuts the 𝑦-axis here at zero. So it’s gonna be plus zero. And every time we increase our 𝑥-coordinate by one — so for example, three to four — the 𝑦-coordinate is also going up by one, so here from three to four. So the equation of this line is 𝑦 equals one 𝑥 plus zero. So the one comes from how much does the 𝑦-coordinate change every time we increase our 𝑥-coordinate by one and the zero comes from where does it cut the 𝑦-axis.
Now here’s another graph. Again cuts the 𝑦-axis at zero, but this time every time we increase our 𝑥-coordinate by one, the 𝑦-coordinate goes up. So if we go from this point to this point on the line, the 𝑦-coordinate has gone up by two. So the slope of the line is two. So the equation of this line is 𝑦 equals two 𝑥 plus zero. In fact we probably wouldn’t even bother writing the plus area; we just write that as 𝑦 equals two 𝑥.
This line again has got an intercept of zero. And now when we increase our 𝑥-coordinate by one, our 𝑦-coordinate is going up by three. So this is 𝑦 equals three 𝑥. What do you think this one is? Well it’s got an intercept of zero, so it’s plus zero on the end. And it’s got a slope or a gradient of positive four, so it’s 𝑦 equals four 𝑥. And this one has an intercept of zero and a slope of five, so it’s 𝑦 equals five 𝑥. So hopefully now you’re getting the feel of this. This value here, the multiplier of 𝑥, tells us how steep that line is. Now for this straight line, again it’s got an intercept of zero. So we’re not to be adding anything onto the equation. But every time I increase my 𝑥-coordinate by one, my 𝑦-coordinate is going down one; it’s negative one. So the slope or the gradient is negative one. So that’s 𝑦 equals negative one 𝑥 plus zero or just 𝑦 equals negative 𝑥.
So this line then intercepts at zero. Every time I increase my 𝑥-coordinate by one, my 𝑦-coordinate goes down by two. So what do you think this equation will be? That’s it, 𝑦 equals minus two 𝑥 and you can put plus zero if you want to. In this case I got an intercept of zero again; so it’s plus zero. And now my 𝑦-coordinate decreases by three every time my 𝑥-coordinate increases by one; so that’s 𝑦 equals minus three 𝑥. And here’s 𝑦 equals minus four 𝑥 and 𝑦 equals minus five 𝑥. So negative slopes or a negative multiple of 𝑥 and an equation of a straight line is gonna be a downhill line from top left down to bottom right. And the size of that number tells us how steeply that line is going to go down.
Now before we go on. Let’s go back to the 𝑦 equals 𝑥 line. Now we’ve looked at positive values; we’ve looked at negative values. And but what if there’re fractional values? Let’s have a look at that. Well if 𝑦 equaled half 𝑥, this is what that would look like. Every time I increase my 𝑥-coordinate by one, my 𝑦-coordinate would increase by half. Now that’s difficult to tell sometimes. But if we look very closely, if I went across two squares, it goes up one whole square. That must mean every time I go across one is going up exactly half a square.
And here is 𝑦 equals a fifth 𝑥. So every time I go across one unit, I go up a fifth of the unit. Again that’s not very easy to tell. But you can see if I start off here five squares later — if my 𝑥-coordinate increases by five — I’ve gone up one whole square. So for every five I go along, I’m going up one. That means I’m going about fifth a square at a time; my slope is positive a fifth.
And what do we think this line is then? Well it takes three squares across, so increasing my 𝑥-coordinate by three for me to go down one whole one. So every time I go across one and I increase my 𝑥 coordinate by one, I must be decreasing my 𝑦-coordinate by a third. So the gradient of this one is going to be negative one third of 𝑥. So the multiplier of 𝑥 is the slope. Whatever number I multiply that by determines the steepness of this line and whether it slopes upwards like this or downwards like this.
Let’s just play around with the other thing, the thing that we’re adding onto the end of that equation — our intercept. So this is 𝑦 equals 𝑥. This line is parallel to that because every time that I increase my 𝑥-coordinate by one, my 𝑦-coordinate also goes up by one. But in this case I’m- I’ve got an intercept of two, so I have to add two onto that equation. So we got the same slope as one, but this time our intercept is two, positive two. There’s blue line then. It’s got the same slope. So it still got that gradient of one. Every time I increase my 𝑥-coordinate by one, my 𝑦-my 𝑦-coordinate also increases by one. But now I’ve got an intercept of four, so this is gonna be 𝑦 equals 𝑥 plus four.
And this line also has a gradient of one because every time I increase my 𝑥-coordinate by one, my 𝑦-coordinate also increases by one, but my intercept is negative six. So my equation is gonna be 𝑦 equals one times 𝑥 or just 𝑦 equals 𝑥 take away six.
Okay here’s one for you to quickly test yourself then. What’s the equation of this line? Think about the slope; think about where it cuts the 𝑦-axis. Well every time I increase my 𝑥-coordinate by one, the 𝑦-coordinate increases by two. And that happens wherever on the line we look. And also it cuts the 𝑦-axis at negative three, so the intercept is negative three. So what’s that gonna make the equation of our line? Remember the slope is the multiple of 𝑥 and the intercept is the other number on the end, in this case negative three.
Now we’ve got this one. Try to work out what the equation of this line is. Well it cuts the 𝑦-axis at four, positive four, so the intercept is positive four. And if I increase my 𝑥-coordinate by two, I decrease my 𝑦-coordinate by one. Now that means that every time I increase my 𝑥-coordinate by one, which is a crucial thing, my 𝑦-coordinate must be going down a half. So the slope is negative a half. So that’s negative a half 𝑥 and the intercept was four, so that’s positive four. 𝑦 equals negative a half 𝑥 plus four.
So let’s do this question the other way round then. We want to draw the graph of 𝑦 equals three 𝑥 take away five. See if you can work out what that graph would look like given you got the slope and the intercept that you have in this equation. Well the intercept is negative five, so it’s gonna cut the 𝑦-axis here. And the slope is three, positive three, So every time I increase my 𝑥-coordinate by one, my 𝑦-coordinate is gonna go up by three. So there’s one point: every time I increase my 𝑥-coordinate by one, my 𝑦-coordinate goes up by three, so there’s another point. And we can keep on doing this: increase the 𝑥 by one, increase the 𝑦 by three, increase the 𝑥 by one, increase the 𝑦 by three.
Now let’s think about what’s gonna happen to the left of the 𝑦-axis well. If every time I increase my 𝑥-coordinate by one, my 𝑦-coordinate goes up by three. If I decrease my 𝑥-coordinate by one, the 𝑦-coordinate is gonna go down by three. So decrease by one, go down by three. And that’s what my graph looks like: 𝑦 equals three 𝑥 minus five.
So let’s just summarise what we’ve learnt so far. The equation of a straight line, it’s always in this format 𝑦 equals some number times 𝑥 plus or minus another number. Now some of you might have heard of that as described as 𝑦 equals 𝑥 plus 𝑏. Some might have heard of it as 𝑦 equals 𝑚𝑥 plus 𝑐. In fact you might have heard of it in different ways. But the slope or gradient tells us whether it’s an uphill line or a downhill line and how steep that line is. And the intercept or the 𝑐 value where it cuts the 𝑦-axis tells us where it cuts the 𝑦-axis. And remember if that’s a positive slope, it goes uphill like this; if it’s a negative slope, it goes downhill like this. And the definition of the slope is by how much does the 𝑦-coordinate change if I increase the 𝑥-coordinate by one along the graph.
Okay one final question then before we go. This graph represents a taxi fare where the 𝑦-coordinate tells you the fare in dollars and the 𝑥-coordinate tells you the number of miles that you’ve travelled. So I want you to pause the video and then answer these three questions: How much is the fixed cost for every journey for getting in the taxi? How much per mile do they charge? And what is the equation of this line?
Well when we travel a distance of zero, the corresponding 𝑦-coordinate would be seven. So they’re charging seven dollars for going no miles. That’s the fixed cost for every journey. How much per mile do they charge? Well every time I increase my 𝑥-coordinate by one and add another mile, the 𝑦-coordinate goes up by two. So that’s two dollars per mile. And what’s the equation of this line? Well the intercept we just worked out was seven, positive seven and the slope was positive two. With an intercept of seven and a slope of two, that’s 𝑦 equals two 𝑥 plus seven.