### Video Transcript

Find the equation of the straight line represented by the graph below in the form of ๐ฆ equals ๐๐ฅ plus ๐.

So first of all, when solving this problem, we actually want to look at what form the equation needs to be left in. And that is ๐ฆ equals ๐๐ฅ plus ๐. Well, if weโre actually using the form ๐ฆ equals ๐๐ฅ plus ๐, so this is actually the general form for the equation of a straight line, then the ๐ is actually our slope and the ๐ is our ๐ฆ-intercept. And what this means, this is actually where it crosses the ๐ฆ-axis. So in order to actually find the equation of our line, what weโre gonna start with is actually the slope, and find the slope for the line that weโve got.

Well, to actually help us find the slope, we have a formula. And that says that the slope, so ๐, is equal to ๐ฆ two minus ๐ฆ one over ๐ฅ two minus ๐ฅ one. And these actually are the coordinates of two points on the line. This can also be thought of in another way. And that is the change in ๐ฆ divided by the change in ๐ฅ. Okay, great. So letโs use this to actually find the slope of our line.

Well, first of all, what Iโve done is actually identify the two points that weโve got on our line. We could actually use any two points. But these are good ones to choose because actually theyโre on definite values. So we have negative two, seven and one, negative one. And as I said, you can pick them any point on the line because actually the slope of a line, if itโs a straight line, wonโt change.

So then next, what Iโve done is Iโve actually labeled these points. So Iโve got ๐ฅ one, ๐ฆ one and ๐ฅ two, ๐ฆ two. And Iโve just done this so you can actually see whatโs gonna go into our formula to find the slope. So therefore, we can say that ๐, so our slope, is gonna be equal to negative one, because that was our ๐ฆ two, minus seven. And thatโs because that was our ๐ฆ one. Then, this is divided by one, because that was our ๐ฅ two, minus, and then weโve got negative two. Because that was our ๐ฅ one. So therefore, we can say that our slope is gonna be equal to negative eight over three. And thatโs cause negative one minus seven is negative eight. And one minus negative two is three.

So therefore, weโve now found our slope. The next stage is to actually find our ๐ฆ-intercept. However, if we just look on the graph, we can see where it is. We can see where the ๐ฆ-axis has been crossed. However, as the graph isnโt accurate, so we donโt actually have a smaller scale on our axes. We canโt tell exactly what that ๐ฆ-intercept is going to be. So weโre gonna have to find it using another way.

Well, if we actually rewrite our ๐ฆ equals ๐๐ฅ plus ๐, but this time substitute in our ๐-value, weโre gonna get ๐ฆ is equal to negative eight over three ๐ฅ plus ๐. So now, to actually find ๐, what we can actually do is substitute in the values for one of the points we know. So Iโm gonna start with the point one, negative one. Because we actually have that as one of the points on our line. So therefore, when we actually substitute that in, weโre gonna get negative one, because that was our ๐ฆ-value, is equal to negative eight over three multiplied by one, cause that was our ๐ฅ-value, plus ๐. So then, if we actually add eight over three to each side, weโre gonna get negative one plus eight over three is equal to ๐.

So therefore, we can say that ๐ is gonna be equal to negative one plus two and two-thirds. What Iโve done is I actually changed our fraction into a mixed number. And I did that by actually seeing how many threes go into eight. Well, two. So thatโs gonna be two. And then thereโs two left over. So itโs two and two-thirds. So therefore, we have a value of ๐ thatโs gonna be one and two-thirds or five over three. So what we can do now is actually check back on our graph to see if this is actually sensible.

Well, if we have a look at our graph, well yes, if we see that itโs one and two-thirds of five-thirds, then this actually seems sensible. Because the graph crosses the ๐ฆ-axis actually just below two. So actually, it does look in the correct place. So therefore, if we substitute back in our values for ๐ and ๐, we can say that the equation of the straight line represented by the graph shown is ๐ฆ equals negative eight over three ๐ฅ plus five over three. And this is in the form ๐ฆ equals ๐๐ฅ plus ๐.