Solve the following simultaneous equations: three 𝑥 minus two 𝑦 equals eight and 𝑥 minus 𝑦 equals two.
So in order to solve the simultaneous equations, what we want to do is find a value of 𝑥 and a value of 𝑦 to satisfy both equations. And as they’re both linear equations, we know that we’re gonna have only one value of 𝑥 and one value of 𝑦. So the first thing I’ve done is label our equations. We have equation one and equation two. And I do this because it’s easier then to show what we’re doing in the next steps. So to solve these simultaneous equations, the method I’m going to use is the elimination method. And in order to use the elimination method, what we need is the coefficient of 𝑥 or the coefficient of 𝑦 to be the same. But this isn’t the case at the moment, and that’s because the coefficient of 𝑥 is three in equation one and just one in equation two. And we know that cause there’s no number in front of the 𝑥, so it’s just gonna be one. And the coefficient of 𝑦 is negative two in equation one and just negative one in equation two.
But what you can do is multiply the second equation by two, and we’re gonna do this because what this is going to do is make our coefficient of 𝑦 the same. And when we do this, we get two 𝑥 minus two 𝑦 equals four. And the reason we can do this is because we’re multiplying everything on the left-hand side and the right-hand side by two. So therefore, it will mean that the equation will still be valid. So two multiplied by 𝑥 gave us the two 𝑥. Two multiplied by negative 𝑦 gave us negative two 𝑦. And then two multiplied by two gave us four. And what I’ve also done is I’ve labelled this equation, and this is now equation three. So as I said, we can now see that we’ve got the same coefficient of 𝑦 in equation one and equation three, and that is negative two. So what’s the next stage?
So now to use the elimination method, what we want to do is eliminate one of our variables. And we eliminate the variable that has the same coefficient in each equation. So in this case, it’s going to be 𝑦. And to help us remember how to do that, because we need to either add or subtract the equations from each other, we have these two little sayings: “Same sign subtract so SSS. And DSA, or different sign add.” Well we can see that the coefficients of 𝑦 in both case have the same sign, because in our equations they’re both negative cause we have negative two and negative two. So therefore, we’re gonna use same sign subtract. And that means we’re gonna subtract one of the equations from the other.
It’s worth noting at this point a common mistake and making sure that you don’t fall into this trap. In orange, I’ve drawn another pair of simultaneous equations. Here we can see that our coefficients that are the same are our 𝑥 coefficient cause we have a coefficient of two at each of our 𝑥 terms. However, the signs in between our 𝑥 and 𝑦 terms are negative and positive. So they’re different. And often students will see different signs in the middle and go, “Well, this must be different sign add.” But this isn’t the case because these aren’t signs that correspond to our coefficients of 𝑥 which are the same. Because if we look at our coefficients of 𝑥, these are both positive. So here we’d also use same sign subtract. It’s just worth noting because as I said it is a common mistake.
So now what we’re gonna do is we’re gonna do equation one minus equation three. I’ve done it that way round because we have more 𝑥 terms in our first equation. So this way it’ll stop us dealing with negatives. So when we do that, we do three 𝑥 minus two 𝑥, which gives us 𝑥. And then we have negative two 𝑦 minus negative two 𝑦. Well if you minus a negative, it’s add. So if you have negative two 𝑦 add two 𝑦, it’s zero. So they eliminate each other, which is what we wanted. And then on the right-hand side of the equation, we’ve got eight minus four, which is just going to be four. So we can say that 𝑥 is equal to four. So great! We’ve found our 𝑥 value. Now what we want to do is find our 𝑦 value.
So now what we want to do is we want to find our 𝑦 value. And to find our 𝑦 value, what we’re gonna do is substitute 𝑥 equals four into one of our equations. It doesn’t matter which one we substitute it into: one, two, or three. But I’ve just decided to substitute it into two. And when we do that, we get four minus 𝑦, and that’s because 𝑥 is equal to four, is equal to two. So then the next step is going to be to add 𝑦 to each side of the equation. I’ve done this because I want positive 𝑦. So when we do that, we get four is equal to two plus 𝑦. And then to find 𝑦, what we do is we subtract two from each side of the equation. When we do that, we get two is equal to 𝑦. So therefore, we can say that the solution to the simultaneous equations three 𝑥 minus two 𝑦 equals eight and 𝑥 minus 𝑦 equals two are 𝑥 equals four and 𝑦 equals two.