### Video Transcript

In the system of equations ππ₯
plus ππ¦ equals four, 38π₯ plus 19π¦ equals 76, π and π are constants. If the system has infinitely many
solutions, what is the value of π minus π?

Now the key here to this question
is the fact that the system has infinitely many solutions. And a system has infinitely many
solutions if both of the equations in the system can be canceled down to the same
equation. So, for example, Iβve got a simple
one here. Weβve got π₯ plus π¦ is equal to
six and two π₯ plus two π¦ is equal to 12. Because as you can see, if I
simplify the second equation by dividing through by two, then Iβm gonna get π₯ plus
π¦ equals six, which is the same as our first equation.

So therefore, if I drew the graph
of both of these, so if I drew the graph of π₯ plus π¦ equals six and two π₯ plus
two π¦ equals 12, then they would overlap completely. And if they overlap completely,
that means that the number in sections would be infinite. So theyβd have infinitely many
solutions.

Okay, so now letβs use this to
solve our problem. So from our definition of a system
that has infinitely many solutions, we can see that what we want to do is find a way
of simplifying the second equation to make the first equation. So what will we do?

Well, first of all, we can see how
many fours go into 76. So if we do 76 divided by four,
this is equal to 19. And we couldβve calculated this
using the bus stop method. So we see how many fours go into
seven, which is one remainder three. Then we see how many fours go into
36, which is nine. So we get the answer, 19. Okay, great, so what do we do with
this?

Well, this tells us that if we want
to get from the second equation and simplify down to the first equation, weβre gonna
divide each of the terms by 19, because as weβve already said, 76 divided by 19 is
four. And when we do that, we get two π₯
β and thatβs because 38 divided by 19 is two β plus π¦ or one π¦ β and thatβs
because 19 divided by 19 is one. But we donβt need to write the
one. So we just write π¦. And then 76 divided by 19 weβve
already shown is four. So we get two π₯ plus π¦ equals
four.

So therefore, we can say that π is
gonna be equal to two and π is gonna be equal to one, which Iβve just added. Cause as I said one π¦, we donβt
usually write the one. But Iβm gonna write it here so we
can see what the value of π is. So now we found the value of π and
π.

We havenβt finished the question
because the question says, βWhat is the value of π minus π?β Well, π minus π is gonna be equal
to two minus one. So therefore, we can say that π
minus π is gonna be equal to one. So that would be the value of π
minus π if π and π are constants and if the system has infinitely many
solutions.