Video: Creating and Solving Simultaneous Equations from Word Problems

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 8 questions incorrectly, and he scored 44 points. Emma answered 14 questions correctly and 6 questions incorrectly, and she scored 58 points. How many points were deducted for each incorrect answer?

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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.

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