### Video Transcript

Plot Linear Equations

We know that every single linear equation is in the form ๐ฆ equals ๐๐ฅ plus ๐, where ๐ is the gradient and ๐ is the ๐ฆ-intercept or where it cuts the ๐ฆ-axis โ cept meaning to cut. Now in this video weโre just going to be able to plot graphs and get them into the form ๐ฆ equals ๐๐ฅ plus ๐ to be able to plot them. So if we want to plot the graph ๐ฆ equals three ๐ฅ minus one or any graph for that matter, the first thing we need is a table of values. Well you have to choose at least three ๐ฅ-values. So we are choosing at least three ๐ฅ-coordinates. And the reason you choose three is basically just in case we get something wrong. So if you choose two and they give you those two points, we could say okay Iโll draw a line in between them. And here we are, hereโs my graph. But actually what happens if we needed the third one. It was here and that tells you either the first, the second, or the third point is wrong. So by doing at least three, it gives us a way of checking what points actually correct into our calculations. Now I like to choose negative one, zero, and one because it involves the least calculation, but you can choose whichever points you like.

So if we do choose negative one, zero, and one, what we need to do is substitute each of those ๐ฅ-coordinates into our function to find the ๐ฆ. So in the first case, weโre going to do ๐ฆ equals three multiplied by negative one minus one. Well, three multiplied by negative one is negative three. And then subtracting one, we get negative four. So our first ๐ฆ-coordinate is negative four. And substituting our next ๐ฅ-coordinate of zero, weโve got three multiplied by zero, which we know is zero. And subtracting one, so zero minus one is minus one. So our next ๐ฆ-coordinate is negative one. And then finally three multiplied by one minus one, so three multiplied by one is three and minus one is equal to two.

So now we need a set of axes to be able to plot our points. Now we can see that our table of value leads us to coordinates to be able to plot. For example, the first one, weโve got negative one as the ๐ฅ-coordinate and then negative four as the ๐ฆ. And the next one is zero as the ๐ฅ and negative one as the ๐ฆ. And then finally, weโve got one in the ๐ฅ-coordinate and two in the ๐ฆ. Now plotting each of these, negative one, negative four, then zero, negative one, and then one, two. And then joining them up, we should get something like this that I prepared earlier.

Now in our next example, we havenโt got it in the form ๐ฆ equals ๐๐ฅ plus ๐. So weโre going to have to do that first. Plot the graph four ๐ฅ plus two ๐ฆ equals ten. So as we can see that as I just said itโs not in the form ๐ฆ equals ๐๐ฅ plus ๐, and it must do- first do a table of values. So first we want it to be in that form. So basically what we need to do is rearrange this to get ๐ฆ as the subject. So what weโre gonna do first is subtract four ๐ฅ from both sides. And that will give us just two ๐ฆ on the left and then ten minus four ๐ฅ on the right. And then this is two multiplied by ๐ฆ or the opposite of multiplied or times by is divide by, so you must now divide both sides by two. Be careful as we need to divide every single term by two. On the left-hand side, we will just have ๐ฆ now. Now weโve got five minus two ๐ฅ.

Now although this isnโt in the form exactly โ๐ฆ equals ๐๐ฅ plus ๐,โ weโve got the ๐ฆ by itself and weโve got the ๐๐ฅ and the ๐ on the same side. So as long as itโs roughly in the same form like that, then youโre okay. So now weโve got this; we can do a table of values. And again Iโm gonna choose the same three coordinates as we did last time, so negative one, zero, and one. And substituting them each individually into our function, five minus two multiplied by negative one. So the two negatives will cancel out giving us a positive two. So the five plus two, and we know the answer to five plus two is seven.

And then for the next case where ๐ฅ is equal to zero, so five minus two times zero, well two times zero is zero. So itโs five minus zero, which we all know is just five. And then for the last ๐ฅ-coordinate of one, weโll substitute that into the function. So weโve got five minus two times one. Two times one is two; so itโs five minus two and five minus two is three. So there we have our table of values and we know that the table of values leads us to coordinates: the first one being negative one, seven, then zero, five, and one, three.

Finally, we just need to plot the points and join them up with a line. So looking for negative one, seven will be minus [minus one] in the ๐ฅ and seven in the ๐ฆ. Then zero, five, zero in the ๐ฅ, five in the ๐ฆ, and then one, three, so across one and up three. And then joining them up, it gives us our straight line here. So we can see it goes through five on the ๐ฆ-axis and it goes down. And this down is related to whatโs in front of the two, being a negative. So this goes on to gradients. So there we have it. To plot linear graphs, what we need is a table of values. But before itโs in a table of values, we must rearrange it to the form ๐ฆ equals ๐๐ฅ plus ๐. Now once we have our table of values, we have some coordinates. Obviously weโve chosen the ๐ฅ-coordinates, but the ๐ฆ-coordinates weโve got from substituting the ๐ฅ-coordinates into the functions individually. And then these give us the coordinates to plot the points on the graph. And once weโve plotted the points, we get a straight edge or a ruler which I havenโt done here and draw a nice line through all these points, giving us the function.