### Video Transcript

In this video, weโre gonna learn how to use the equation for a straight line to find the slope and the ๐ฆ-intercept of that line. The general format of an equation of the straight line is ๐ฆ equals ๐๐ฅ plus ๐ and you might know this as ๐ฆ equals ๐๐ฅ plus ๐ or ๐ฆ equals ๐๐ฅ plus ๐. That doesnโt really matter. The point is that the number that youโre multiplying ๐ฅ by tells you the slope of the line and the number thatโs on its own, added or subtracted at the end, tells you the ๐ฆ-intercept of the line.

So, for example, if we had the equation ๐ฆ equals three ๐ฅ plus two, three would be the slope and positive two would be the ๐ฆ-intercept. But what does that actually mean? Well, put simply, the ๐ฆ-intercept is where it cuts the ๐ฆ-axis. So thatโs here. So another way of thinking about that is if we look at this point here, the co- the ๐ฅ-coordinate is zero and the ๐ฆ-coordinate is two. So the ๐ฆ-coordinate when the ๐ฅ-coordinate is zero is the ๐ฆ-intercept. And if we think about that in terms of our equation, if I put a value of zero into my equation, for ๐ฅ, Iโve got ๐ฆ equals three times zero plus two or three times zero is just zero. So ๐ฆ equals zero plus two; ๐ฆ is equal to two.

So weโve got a few ways to think about the ๐ฆ-intercept. Firstly, itโs just this number in our equation. Secondly, itโs just where does it cut the ๐ฆ-axis. Or thirdly, itโs what answer do I get for ๐ฆ when I plug in the ๐ฅ-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 ๐ฅ-coordinate by one, the ๐ฆ-coordinate goes up by three, because thatโs positive three. So, gonna increase my ๐ฅ-coordinate by one; the ๐ฆ-coordinate goes up by three. If I increase my ๐ฅ-coordinate by one again, the ๐ฆ-coordinate goes up by three, which of course means, if I go backwards, if I decrease my ๐ฅ-coordinate by one, the ๐ฆ-coordinateโs 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 weโve got our equation in the ๐ฆ equals ๐๐ฅ plus ๐ format, the slope is the multiplier of ๐ฅ or the coefficient of ๐ฅ in that equation.

So letโs look at a few examples then. ๐ฆ equals negative five ๐ฅ plus two. Itโs in the ๐ฆ equals ๐๐ฅ plus ๐ format, so the slope is just negative five and the ๐ฆ-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 ๐ฅ-coordinate, what changes there in the ๐ฆ-coordinate to get back to the line? In this case, thatโs negative five. And the ๐ฆ-intercept, remember, is just where does it cut the ๐ฆ-axis; so, in this case, that is two.

Right. Next example: ๐ฆ equals ๐ฅ minus a half. Well, the slope is the thing that weโve multiplied ๐ฅ by, and the intercept is the thing thatโs just added on at the end. So weโve got this equation in basically the right format ๐ฆ equals something times ๐ฅ plus another number, except we havenโt written that number in front of the ๐ฅ, which means itโs one. When itโs one times ๐ฅ, we donโt normally write the one in that, but it is really there. And the other slight difference from normal is that although we say itโs ๐๐ฅ plus ๐, the number that weโve added on here is negative a half. So here, the slope is one and the ๐ฆ-intercept is negative a half. That means it cuts the ๐ฆ-axis when ๐ฆ is negative a half and every time I increase my ๐ฅ-coordinate by one, the corresponding ๐ฆ-coordinate, on that equation, goes up by one as well. So in these questions, you wouldnโt necessarily have to draw the graph. But just out of interest, thatโs what it would look like in this case. Every time I increase my ๐ฅ-coordinate by one, the corresponding ๐ฆ-coordinate goes up by one when I move back to the line. And the ๐ฆ-intercept, where it cuts the ๐ฆ-axis, is negative a half.

How about this one: ๐ฆ equals seven ๐ฅ. Well, weโve got a multiplier of ๐ฅ, weโve got just one ๐ฆ on its own, so the slope is positive seven. But what have we added on to the end? Well, weโve added on nothing. So that means our ๐ฆ-intercept is zero; it goes through the origin. The line ๐ฆ equals seven ๐ฅ has a coordinate; when ๐ฅ is zero, the ๐ฆ-coordinate is zero as well.

Now sometimes you donโt get the equation in the right format and you have to do a little bit of work in order to get it there. This isnโt in the ๐ฆ equals ๐๐ฅ plus ๐ format, weโve got ๐ฆ plus two ๐ฅ equals negative seven. So, we can use inverse operations to try and clear off everything else from the side of the equation thatโs got the ๐ฆ on it and see what we end up with. So on the left-hand side, weโve got ๐ฆ plus two ๐ฅ. Well the inverse operation of adding two ๐ฅ is subtracting two ๐ฅ. So Iโm gonna subtract two ๐ฅ from both sides of my equation. And on the left-hand side, ๐ฆ plus two ๐ฅ minus two ๐ฅ just leaves me with ๐ฆ. Well that was the point of doing that. And on the right-hand side, if I take away two ๐ฅ, I get negative seven minus two ๐ฅ. Now, Iโve got two terms here. Iโve got a negative seven and Iโve got a negative two ๐ฅ. Now it doesnโt matter whether I start off at negative seven and then I take away two ๐ฅ from that or I start off at negative two ๐ฅ 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 ๐ฆ equals ๐๐ฅ plus ๐ format, thatโs gonna be easier for you to work out what your slope and ๐ฆ-intercepts are. The coefficient of ๐ฅ, or the multiplier of ๐ฅ, 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 ๐ฆ-intercept is negative seven, so itโs gonna cut the ๐ฆ-axis at negative seven. And if youโre interested in the graph, look it cuts the ๐ฆ-axis here at negative seven and every time I increase the ๐ฅ-coordinate by one, the ๐ฆ-coordinate decreases by two along that line. So the slope is negative two.

Now in our next example, two ๐ฆ plus three ๐ฅ equals six. Itโs still the equation of a straight line, but itโs not quite arranged in the right format ๐ฆ equals ๐๐ฅ plus ๐. So, again, Iโm gonna look for inverse operations. Iโve got two ๐ฆ plus three ๐ฅ. So, the opposite of adding three ๐ฅ is subtracting three ๐ฅ. Iโm gonna do that to both sides of my equation. And on the left-hand side, Iโve got two ๐ฆ plus three ๐ฅ minus three ๐ฅ is just two ๐ฆ; and on the right-hand side, Iโve got six minus three ๐ฅ. But again, Iโm gonna swap those two around because Iโd rather have it in the ๐๐ฅ plus ๐ format, and they are both equivalent. So you do need to be a little bit careful with the signs there. So weโve got two ๐ฆ is equal to negative three ๐ฅ plus six. But weโre not quite ready to go yet. Look, we still got this two ๐ฆ here. So, ๐ฆ times two. Now the inverse operation of times-ing by two is dividing by two, so if I wanna just get ๐ฆ 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 ๐ฆ is one ๐ฆ, a half of negative three is negative one and a half, but in fact Iโm just gonna write it in this format: negative three divided by two. It actually saves doing any work and thatโs a perfectly acceptable form of the value there, and then a half of six is three, so plus three. Now weโve got it in the right format: ๐ฆ is equal to something times ๐ฅ plus something. And the something times ๐ฅ 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 ๐ฆ-intercept. So thatโs where it cuts the ๐ฆ-axis, and the slope tells us that itโs a downhill. Every time I increase my ๐ฅ-coordinate by one and move to the right by one, then the-the ๐ฆ-coordinate goes down by one and half.

So just to summarise what weโve learned then; we can get equations in the format ๐ฆ equals something times ๐ฅ plus something. Now the multiple of ๐ฅ, the coefficient of ๐ฅ 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 ๐ฆ-axis. So weโve got an example here: this straight line cuts the ๐ฆ-axis at two, so the ๐ฆ-intercept is two, positive two. And moving along that line, every time I increase the ๐ฅ-coordinate by one, the corresponding ๐ฆ-coordinate also goes up by one. So, the slope is one, positive one. So feeding that back into our equation, in this case, ๐ would be equal to one and ๐ would be equal to positive two. And the equation then of our line is: ๐ฆ equals one ๐ฅ plus two, but of course remember we donโt normally bother writing the one in there, so we could rub that out; ๐ฆ equals ๐ฅ 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 ๐ฆ-intercept are.