### Video Transcript

In a test with 20 questions, 𝑥
marks are awarded for each correct answer and 𝑦 marks are deducted for each
incorrect answer. Benjamin answered 12 questions
correctly and eight questions incorrectly, and he scored 44 points. Emma answered 14 questions
correctly and six questions incorrectly, and she scored 58 points. How many points were deducted for
each incorrect answer?

So in this question, we can see
that we’ve got two variables. So what we’re gonna want to do is
use the information we’ve been given to set up two equations and solve them
simultaneously. And then what we’re looking to find
is what our 𝑦 is because we want to find how many points were deducted for each
incorrect answer. Well, the first bit of information
we can use to form the first equation is the fact that Benjamin answered 12
questions correctly and eight questions incorrectly and he scored 44 points.

So first of all, we’re gonna begin
with 12𝑥, and that’s because we know that there are 𝑥 marks for each correct
answer and Benjamin answered 12 questions correctly. And then this is gonna be minus
eight 𝑦. And we get that because we’re told
that 𝑦 marks are deducted for each incorrect answer. So therefore, we get eight 𝑦
because there’re eight questions incorrectly answered by Benjamin. However, the reason it’s minus
eight 𝑦 is it’s because 𝑦 marks are deducted. So we’re taking them away. So now, we’ve got an expression
12𝑥 minus eight 𝑦. And then this is equal to 44 cause
we’re told that Benjamin scored 44 points overall. Okay, great! So I’ve labeled this equation one
cause this will help us when we’re identifying what to do in the next steps.

Now, let’s look at equation
two. Well for equation two, we’re gonna
have 14𝑥 minus six 𝑦 equals 58. And that’s cause we’re told that
Emma answered 14 questions correctly, so that’s 14𝑥, and six questions incorrectly,
so we subtract six 𝑦. And then the total score that she
got was 58 points. Okay, great! So now what’s our next step? So what we want to do now is solve
our equations simultaneously. And the method we’re gonna use is
elimination. However, to use elimination, what
we need to do is we need to eliminate either 𝑥 or 𝑦. And to do that, we have to have the
same coefficient of either 𝑥 or 𝑦. It doesn’t matter if the signs
aren’t the same, it just has to be the same value in front of our 𝑥 or 𝑦.

So this isn’t the case with our
equations. So therefore, what makes this a
slightly more complex problem is the fact that we now need to find a number that we
need to multiply each of our equations by to enable us to have the same coefficient
of either 𝑥 or 𝑦. Now, what we can do is we can
multiply equation one by three and equation two by four because what this is going
to do is give us a coefficient in both cases of 24 for our 𝑦. Well, in fact, it’ll be negative
24. And that’s because eight multiplied
by three is 24 and six multiplied by four is also 24.

So we’re gonna start with equation
one. And if we multiply this by three,
what we’re gonna get is 36𝑥 minus 24𝑦 equals 132. I’m gonna call this equation
three. And then if we take a look at
equation two, if we multiply this by four, we’re gonna get 56𝑥 minus 24𝑦 equals
232. And we’ve called this equation
four. And it’s worth noting at this
point, be careful of a common mistake. And that is where people forget to
multiply each of the terms. And most likely, it’s the term on
the right-hand side of our equation, so the numerical value they forget to
multiply.

Well, now, if we take a look at the
coefficients we’ve got, we’ve got 24𝑦 in both of our equations. And in fact, we also have them both
as negative, so they have the same signs. So to eliminate the values, what
we’re gonna do is same-sign subtract. And that’s cause if we have
negative 24 minus negative 24 is the same as negative 24 add 24, which is just
zero.

However, it’s worth mentioning here
a common mistake cause people often just look at the central signs of the equations
and then deal with those. So if in this case, they’re both
the same. However, if you’re trying to
eliminate the 𝑥-terms, then it would in fact be the signs associated with the
coefficient of our 𝑥-terms which will be the ones that you would look at. So now, to enable us to eliminate
one of our variables, so in this case 𝑦, what we’re gonna do is subtract equation
three from equation four. And when we do that, what we get is
20𝑥 is equal to 100. And that’s because 56𝑥 minus 36𝑥
is 20𝑥. And then, as we said, we already
eliminate the 𝑦-terms. And then we’ve got 232 minus 132,
which is 100. So then we divide through by 20,
what we get is 𝑥 is equal to five.

So great, we found out what 𝑥
is. However, this is not what the
question is looking for cause it wants us to find out how many points were deducted
for each incorrect answer. And this is the variable 𝑦. So then, to find 𝑦, what we do is
substitute 𝑥 equals five into equation one. We could choose any equation. I just happen to have chosen
equation one here. So when we do that, what we’re
gonna get is 12 multiplied by five minus eight 𝑦 equals 44, which is gonna give us
60 minus eight 𝑦 equals 44. So then, what we can do is
rearrange to solve to find 𝑦. And so what we do is subtract 44
from each side of the equation and add eight 𝑦. And we do this so that we have a
positive 𝑦-term.

So then what we get is 16 is equal
to eight 𝑦. So then if we divide through by
eight, we’re gonna get two is equal to 𝑦. So therefore, as we found out that
𝑦 is equal to two, we can answer the question cause we can say that two points were
deducted for each incorrect answer. And it would be possible to check
our answer by substituting our 𝑥- and 𝑦-values into any one of our four
equations.