### Video Transcript

If the two straight lines two ๐ฅ plus five ๐ฆ equals 10 and ๐๐ฅ plus ๐ฆ equals five are parallel, which of the following graphs represents the second line?

And then weโre given five graphs to choose from. In order to help us solve this problem, letโs begin by recalling what we know about parallel lines. If these two straight lines are parallel, they will have the same slope or the same gradient. And so in order to work out which of the graphs represents the second line, weโre going to begin by finding the slope of the first line.

Now, at the moment, itโs not in a particularly useful form. So weโre going to rearrange it into slopeโintercept form, that is, of the form ๐ฆ equals ๐๐ฅ plus ๐. When itโs in this form, the value of ๐, which is the coefficient of ๐ฅ, represents the slope of the line, whereas the value of ๐ tells us the location of the ๐ฆ-intercept.

So, for the equation two ๐ฅ plus five ๐ฆ equals 10, we need to make ๐ฆ the subject. Weโll begin by subtracting two ๐ฅ from both sides. That gives us five ๐ฆ equals 10 minus two ๐ฅ. Then, we divide through by five. And of course we can individually divide each term on the right-hand side by five. 10 divided by five is two, and negative two ๐ฅ divided by five is negative two ๐ฅ over five, or negative two-fifths ๐ฅ. And so the slope of our first line is negative two-fifths. But of course since theyโre parallel, we know that the slope of our second line is also negative two-fifths.

Now, if the slope is negative, if itโs less than zero, this means that, from left to right, the line slopes downwards. Thatโs really useful because it means we can instantly disregard options (C) and (E). So, what next?

Well, letโs take the value of the slope and use it to find the value of ๐ in the equation of our second line. This will allow us to find the value of the ๐ฆ-intercept. And actually, we have three different graphs with three different ๐ฆ-intercepts. So we have the equation ๐๐ฅ plus ๐ฆ equals five. Letโs subtract ๐๐ฅ from both sides just to get it into slopeโintercept form, giving us ๐ฆ equals five minus ๐๐ฅ.

Now, the coefficient of ๐ฆ is one, so we donโt need to go any further. Since we said the slope of this second line is also negative two-fifths, the coefficient of ๐ฅ must be negative two-fifths. So negative ๐ must be equal to negative two-fifths, which means ๐ is equal to two-fifths.

Now, in fact, we see that when we put this value of ๐ into the expression ๐ฆ equals five minus ๐๐ฅ, it wasnโt actually necessary to do so to be able to work out the value of the ๐ฆ-intercept. The value of the ๐ฆ-intercept is the constant when the expression is of the form ๐ฆ equals ๐๐ฅ plus ๐. So the ๐ฆ-intercept must be equal to five. The only graph that satisfies this criteria is graph (D).

But of course itโs not enough just to find the gradient and the ๐ฆ-intercept. We do in fact need to check that this line has a gradient of negative two-fifths or, alternatively, that a second point on the line satisfies the equation ๐ฆ equals five minus two ๐ฅ.

Letโs use the second method just to verify our answer. This passes through the point with coordinates five, three. Knowing that the equation is ๐ฆ equals five minus two-fifths ๐ฅ, weโll substitute ๐ฅ equals five in. And we hope that weโll get ๐ฆ equals three back out. That gives us ๐ฆ equals five minus two-fifths times five. We can then cancel by dividing through by five. So ๐ฆ is five minus two, which is three. This means the ordered pair satisfies the equation ๐ฆ equals five minus two-fifths ๐ฅ. And this verifies to us that graph (D) is in fact the graph that we were after.