Video: SAT Practice Test 1 β€’ Section 3 β€’ Question 20

In the system of equations π‘šπ‘₯ + 𝑛𝑦 = 4, 38π‘₯ + 19𝑦 = 76, π‘š and 𝑛 are constants. If the system has infinitely many solutions, what is the value of π‘š βˆ’ 𝑛?

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Video Transcript

In the system of equations π‘šπ‘₯ plus 𝑛𝑦 equals four, 38π‘₯ plus 19𝑦 equals 76, π‘š and 𝑛 are constants. If the system has infinitely many solutions, what is the value of π‘š minus 𝑛?

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, I’ve got a simple one here. We’ve got π‘₯ plus 𝑦 is equal to six and two π‘₯ plus two 𝑦 is equal to 12. Because as you can see, if I simplify the second equation by dividing through by two, then I’m gonna get π‘₯ plus 𝑦 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 𝑦 equals six and two π‘₯ plus two 𝑦 equals 12, then they would overlap completely. And if they overlap completely, that means that the number in sections would be infinite. So they’d have infinitely many solutions.

Okay, so now let’s 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 could’ve 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, we’re gonna divide each of the terms by 19, because as we’ve already said, 76 divided by 19 is four. And when we do that, we get two π‘₯ β€” and that’s because 38 divided by 19 is two β€” plus 𝑦 or one 𝑦 β€” and that’s because 19 divided by 19 is one. But we don’t need to write the one. So we just write 𝑦. And then 76 divided by 19 we’ve already shown is four. So we get two π‘₯ plus 𝑦 equals four.

So therefore, we can say that π‘š is gonna be equal to two and 𝑛 is gonna be equal to one, which I’ve just added. Cause as I said one 𝑦, we don’t usually write the one. But I’m gonna write it here so we can see what the value of 𝑛 is. So now we found the value of π‘š and 𝑛.

We haven’t finished the question because the question says, β€œWhat is the value of π‘š minus 𝑛?” Well, π‘š minus 𝑛 is gonna be equal to two minus one. So therefore, we can say that π‘š minus 𝑛 is gonna be equal to one. So that would be the value of π‘š minus 𝑛 if π‘š and 𝑛 are constants and if the system has infinitely many solutions.

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