Find the equation of the straight line represented by the graph below in the form of 𝑦 equals 𝑚𝑥 plus 𝑐.
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 𝑦 equals 𝑚𝑥 plus 𝑐. Well, if we’re actually using the form 𝑦 equals 𝑚𝑥 plus 𝑐, so this is actually the general form for the equation of a straight line, then the 𝑚 is actually our slope and the 𝑐 is our 𝑦-intercept. And what this means, this is actually where it crosses the 𝑦-axis. So in order to actually find the equation of our line, what we’re gonna start with is actually the slope, and find the slope for the line that we’ve got.
Well, to actually help us find the slope, we have a formula. And that says that the slope, so 𝑚, is equal to 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 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 𝑦 divided by the change in 𝑥. Okay, great. So let’s use this to actually find the slope of our line.
Well, first of all, what I’ve done is actually identify the two points that we’ve got on our line. We could actually use any two points. But these are good ones to choose because actually they’re 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 it’s a straight line, won’t change.
So then next, what I’ve done is I’ve actually labeled these points. So I’ve got 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two. And I’ve just done this so you can actually see what’s gonna go into our formula to find the slope. So therefore, we can say that 𝑚, so our slope, is gonna be equal to negative one, because that was our 𝑦 two, minus seven. And that’s because that was our 𝑦 one. Then, this is divided by one, because that was our 𝑥 two, minus, and then we’ve got negative two. Because that was our 𝑥 one. So therefore, we can say that our slope is gonna be equal to negative eight over three. And that’s cause negative one minus seven is negative eight. And one minus negative two is three.
So therefore, we’ve now found our slope. The next stage is to actually find our 𝑦-intercept. However, if we just look on the graph, we can see where it is. We can see where the 𝑦-axis has been crossed. However, as the graph isn’t accurate, so we don’t actually have a smaller scale on our axes. We can’t tell exactly what that 𝑦-intercept is going to be. So we’re gonna have to find it using another way.
Well, if we actually rewrite our 𝑦 equals 𝑚𝑥 plus 𝑐, but this time substitute in our 𝑚-value, we’re gonna get 𝑦 is equal to negative eight over three 𝑥 plus 𝑐. So now, to actually find 𝑐, what we can actually do is substitute in the values for one of the points we know. So I’m 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, we’re gonna get negative one, because that was our 𝑦-value, is equal to negative eight over three multiplied by one, cause that was our 𝑥-value, plus 𝑐. So then, if we actually add eight over three to each side, we’re gonna get negative one plus eight over three is equal to 𝑐.
So therefore, we can say that 𝑐 is gonna be equal to negative one plus two and two-thirds. What I’ve 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 that’s gonna be two. And then there’s two left over. So it’s two and two-thirds. So therefore, we have a value of 𝑐 that’s 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 it’s one and two-thirds of five-thirds, then this actually seems sensible. Because the graph crosses the 𝑦-axis actually just below two. So actually, it does look in the correct place. So therefore, if we substitute back in our values for 𝑚 and 𝑐, we can say that the equation of the straight line represented by the graph shown is 𝑦 equals negative eight over three 𝑥 plus five over three. And this is in the form 𝑦 equals 𝑚𝑥 plus 𝑐.