Question Video: Identifying Graphs of Linear Equations in Point-Slope Form

Which of the following graphs represents the equation 𝑦 − 5 = (2/3)(𝑥 −3)? [A] Graph A [B] Graph B [C] Graph C [D] Graph D [E] Graph E

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Video Transcript

Which of the following graphs represents the equation 𝑦 minus five equals two-thirds multiplied by 𝑥 minus three?

And we have five graphs: (A), (B), (C), (D), and (E). So what we have is an equation, and what it’s in is the point-slope form. So that’s 𝑦 minus 𝑦 sub one equals 𝑚 multiplied by 𝑥 minus 𝑥 sub one, where 𝑚 is the slope and the point that lies on the straight line is 𝑥 sub one, 𝑦 sub one. So the first thing we can find from our equation is our 𝑚, so our slope, because our 𝑚 is equal to two-thirds.

Well, just reminding ourselves that the slope is equal to the change in 𝑦 divided by the change in 𝑥, so therefore what this means is that for every two units up, the graph goes; it goes three units across. And that’s because as we’ve said the top number is the change in 𝑦 and the bottom number is the change in 𝑥.

So great, we know our slope is two-thirds. And what we can also do now is use our equation to find a point on the line. So what we can see is that we’ve got a point on the line which must be with the coordinates three, five. So we can know that our 𝑥 sub one, 𝑦 sub one is equal to three, five. So great, what we have are both bits of information we need to identify which graph is the correct graph.

So first of all, we’re gonna start with a point on the line. Well, if we take a look at graph (A), we can see that the point three, five does lie on the line. So therefore, this could possibly be the correct graph. Well, if we take a look at graph (B), we can see in fact the point does not lie on the line because three, five, we can see here, is not on the line. So therefore, this cannot be the correct graph.

Taking a look at point (C), we can see that three, five does lie on the line. So this could be the correct graph. If we take a look at graph (D), we can see that, on graph (D), the point doesn’t lie on the line either cause it’s at the very edge of the axes. So this is not the correct graph either.

So what is worth mentioning at this point is a common mistake that can be made. And that’s getting the 𝑥- and 𝑦-coordinates the wrong way round. So we can see here this point is in fact five, three. Then, if we move on to the final graph, graph (E), we could see that this could possibly be the correct graph because the point does lie on the line cause we have the point three, five on our line.

So great, what we’ve done now is ruled out two of our graphs, (B) and (D). So now what we need to do is take a look at the other bit of information we’ve got, and that’s the slope of our graph, to see if this can help us decide which of the graphs that are left is the correct graph. So one thing we know about the shape of the graph is that if we have a positive slope, the line goes up to the right. And if we have a negative slope, it goes down to the right.

Well, if we think about the slope we’ve got, it’s two-thirds, which is positive. So therefore, what we’re looking for is the slope that goes up to the right. Well, this means we can rule out one more graph. We can rule out graph (E) because in fact this is a negative slope. So this cannot be the correct graph.

So great, what we’re left with is two graphs, graph (A) or (C). So now, to determine which one of the graphs, (A) or (C), is the correct graph, we can use our slope in a couple of ways. First of all, we could go and work out the slope of both of the lines separately. However, what we can also do is use what we know this slope means.

So to use this method, what we’ve done is chosen a point on graph (A). And we’ve chosen the point negative three, one. It doesn’t matter which point you choose, just one that’s easy to read off the scale. So we know from our slope, which is two-thirds, that for every two units up, so the change in 𝑦, our line should go three units across. So we can see from graph (A) that if we start on a point on the line, then we get up two units and along three units, we do in fact land at a point again on our straight line. So we can say that the slope is two-thirds.

However, if we take a look at graph (C), and which is a point on our line, if we go up two units and along three units, what this in fact does is takes us beyond our line. So therefore, we can say that the slope is not in fact two-thirds. But in fact, if we did want to find the slope of our line, we can see that if we go up three units and along two units, so here I’ve picked another couple of points, then we lie back on the line. So we could say that the slope of the line would be equal to three over two. So what we can say is that the correct graph is graph (A) because this is the graph that represents the equation 𝑦 minus five equals two-thirds multiplied by 𝑥 minus three.

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