In this video, we’re gonna learn how to use the equation for a straight line to find the slope and the 𝑦-intercept of that line. The general format of an equation of the straight line is 𝑦 equals 𝑚𝑥 plus 𝑏 and you might know this as 𝑦 equals 𝑚𝑥 plus 𝑐 or 𝑦 equals 𝑎𝑥 plus 𝑏. That doesn’t really matter. The point is that the number that you’re multiplying 𝑥 by tells you the slope of the line and the number that’s on its own, added or subtracted at the end, tells you the 𝑦-intercept of the line.
So, for example, if we had the equation 𝑦 equals three 𝑥 plus two, three would be the slope and positive two would be the 𝑦-intercept. But what does that actually mean? Well, put simply, the 𝑦-intercept is where it cuts the 𝑦-axis. So that’s here. So another way of thinking about that is if we look at this point here, the co- the 𝑥-coordinate is zero and the 𝑦-coordinate is two. So the 𝑦-coordinate when the 𝑥-coordinate is zero is the 𝑦-intercept. And if we think about that in terms of our equation, if I put a value of zero into my equation, for 𝑥, I’ve got 𝑦 equals three times zero plus two or three times zero is just zero. So 𝑦 equals zero plus two; 𝑦 is equal to two.
So we’ve got a few ways to think about the 𝑦-intercept. Firstly, it’s just this number in our equation. Secondly, it’s just where does it cut the 𝑦-axis. Or thirdly, it’s what answer do I get for 𝑦 when I plug in the 𝑥-coordinate of zero. And those last two things mean exactly the same thing.
And slope is all about how steeply uphill or downhill the line goes. So in this case, our slope is three. And that means every time I increase my 𝑥-coordinate by one, the 𝑦-coordinate goes up by three, because that’s positive three. So, gonna increase my 𝑥-coordinate by one; the 𝑦-coordinate goes up by three. If I increase my 𝑥-coordinate by one again, the 𝑦-coordinate goes up by three, which of course means, if I go backwards, if I decrease my 𝑥-coordinate by one, the 𝑦-coordinate’s gonna go down by three. So I can work out what my other points are gonna be, and I can draw my straight line. So the slope is not only the steepness of the line we can recognise here. If we’ve got our equation in the 𝑦 equals 𝑚𝑥 plus 𝑏 format, the slope is the multiplier of 𝑥 or the coefficient of 𝑥 in that equation.
So let’s look at a few examples then. 𝑦 equals negative five 𝑥 plus two. It’s in the 𝑦 equals 𝑚𝑥 plus 𝑏 format, so the slope is just negative five and the 𝑦-intercept is positive two. And if we think about that in terms of the graph, if I take a point on that line and then I add one to the 𝑥-coordinate, what changes there in the 𝑦-coordinate to get back to the line? In this case, that’s negative five. And the 𝑦-intercept, remember, is just where does it cut the 𝑦-axis; so, in this case, that is two.
Right. Next example: 𝑦 equals 𝑥 minus a half. Well, the slope is the thing that we’ve multiplied 𝑥 by, and the intercept is the thing that’s just added on at the end. So we’ve got this equation in basically the right format 𝑦 equals something times 𝑥 plus another number, except we haven’t written that number in front of the 𝑥, which means it’s one. When it’s one times 𝑥, we don’t normally write the one in that, but it is really there. And the other slight difference from normal is that although we say it’s 𝑚𝑥 plus 𝑏, the number that we’ve added on here is negative a half. So here, the slope is one and the 𝑦-intercept is negative a half. That means it cuts the 𝑦-axis when 𝑦 is negative a half and every time I increase my 𝑥-coordinate by one, the corresponding 𝑦-coordinate, on that equation, goes up by one as well. So in these questions, you wouldn’t necessarily have to draw the graph. But just out of interest, that’s what it would look like in this case. Every time I increase my 𝑥-coordinate by one, the corresponding 𝑦-coordinate goes up by one when I move back to the line. And the 𝑦-intercept, where it cuts the 𝑦-axis, is negative a half.
How about this one: 𝑦 equals seven 𝑥. Well, we’ve got a multiplier of 𝑥, we’ve got just one 𝑦 on its own, so the slope is positive seven. But what have we added on to the end? Well, we’ve added on nothing. So that means our 𝑦-intercept is zero; it goes through the origin. The line 𝑦 equals seven 𝑥 has a coordinate; when 𝑥 is zero, the 𝑦-coordinate is zero as well.
Now sometimes you don’t get the equation in the right format and you have to do a little bit of work in order to get it there. This isn’t in the 𝑦 equals 𝑚𝑥 plus 𝑏 format, we’ve got 𝑦 plus two 𝑥 equals negative seven. So, we can use inverse operations to try and clear off everything else from the side of the equation that’s got the 𝑦 on it and see what we end up with. So on the left-hand side, we’ve got 𝑦 plus two 𝑥. Well the inverse operation of adding two 𝑥 is subtracting two 𝑥. So I’m gonna subtract two 𝑥 from both sides of my equation. And on the left-hand side, 𝑦 plus two 𝑥 minus two 𝑥 just leaves me with 𝑦. Well that was the point of doing that. And on the right-hand side, if I take away two 𝑥, I get negative seven minus two 𝑥. Now, I’ve got two terms here. I’ve got a negative seven and I’ve got a negative two 𝑥. Now it doesn’t matter whether I start off at negative seven and then I take away two 𝑥 from that or I start off at negative two 𝑥 and I take away seven from that; I get the same thing. But, I would recommend that you write it in this format because then, that is 𝑦 equals 𝑚𝑥 plus 𝑏 format, that’s gonna be easier for you to work out what your slope and 𝑦-intercepts are. The coefficient of 𝑥, or the multiplier of 𝑥, in this case, is negative two. So the slope is negative two. So this is a fairly steep downhill straight line. And the intercept of 𝑦-intercept is negative seven, so it’s gonna cut the 𝑦-axis at negative seven. And if you’re interested in the graph, look it cuts the 𝑦-axis here at negative seven and every time I increase the 𝑥-coordinate by one, the 𝑦-coordinate decreases by two along that line. So the slope is negative two.
Now in our next example, two 𝑦 plus three 𝑥 equals six. It’s still the equation of a straight line, but it’s not quite arranged in the right format 𝑦 equals 𝑚𝑥 plus 𝑏. So, again, I’m gonna look for inverse operations. I’ve got two 𝑦 plus three 𝑥. So, the opposite of adding three 𝑥 is subtracting three 𝑥. I’m gonna do that to both sides of my equation. And on the left-hand side, I’ve got two 𝑦 plus three 𝑥 minus three 𝑥 is just two 𝑦; and on the right-hand side, I’ve got six minus three 𝑥. But again, I’m gonna swap those two around because I’d rather have it in the 𝑚𝑥 plus 𝑏 format, and they are both equivalent. So you do need to be a little bit careful with the signs there. So we’ve got two 𝑦 is equal to negative three 𝑥 plus six. But we’re not quite ready to go yet. Look, we still got this two 𝑦 here. So, 𝑦 times two. Now the inverse operation of times-ing by two is dividing by two, so if I wanna just get 𝑦 on its own, I need to divide both sides of my equation by two. And that means every term on both sides. So going through term by term on the left-hand side, a half of two 𝑦 is one 𝑦, a half of negative three is negative one and a half, but in fact I’m just gonna write it in this format: negative three divided by two. It actually saves doing any work and that’s a perfectly acceptable form of the value there, and then a half of six is three, so plus three. Now we’ve got it in the right format: 𝑦 is equal to something times 𝑥 plus something. And the something times 𝑥 tells us the slopes are negative three over two, or negative one and a half is the slope. And the something on its own at the end, plus three, positive three, is the 𝑦-intercept. So that’s where it cuts the 𝑦-axis, and the slope tells us that it’s a downhill. Every time I increase my 𝑥-coordinate by one and move to the right by one, then the-the 𝑦-coordinate goes down by one and half.
So just to summarise what we’ve learned then; we can get equations in the format 𝑦 equals something times 𝑥 plus something. Now the multiple of 𝑥, the coefficient of 𝑥 is known as the slope of the gradient; and that tells us how steep the line is. And the number on its own that we add tells us where it cuts the 𝑦-axis. So we’ve got an example here: this straight line cuts the 𝑦-axis at two, so the 𝑦-intercept is two, positive two. And moving along that line, every time I increase the 𝑥-coordinate by one, the corresponding 𝑦-coordinate also goes up by one. So, the slope is one, positive one. So feeding that back into our equation, in this case, 𝑚 would be equal to one and 𝑏 would be equal to positive two. And the equation then of our line is: 𝑦 equals one 𝑥 plus two, but of course remember we don’t normally bother writing the one in there, so we could rub that out; 𝑦 equals 𝑥 plus two. And sometimes the equation we were given is not in the right format for us, so we need to use inverse operations to rearrange that so that we can work out what the slope and 𝑦-intercept are.