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.
