Find all of the solutions to the simultaneous equations 𝑥 minus 𝑦 equals six and 𝑥 squared minus nine 𝑥𝑦 plus 𝑦 squared equals 36.
So in this question, we’re looking to solve a pair of simultaneous equations. I’ve labeled them one and two. And I do that because it’s easier to be clear exactly which steps you’re doing when you have the equations labeled.
So in order to solve this simultaneous equation problem, the method we’re gonna use is a method called substitution. And in order to use this method, what we need to do is rearrange one of our equations to make either 𝑥 or 𝑦 the subject and then substitute this back in to the other equation.
The equation we’re gonna use in this question is equation one. And I’ve chosen to make 𝑥 the subject, and we can do either 𝑥 or 𝑦. But as I said, I’ve chosen to do 𝑥 with this problem. So what we do is we add 𝑦 to each side of the equation. And when we do that, we get 𝑥 is equal to six plus 𝑦. So this is our third equation.
So what I’m gonna do now is substitute equation three into equation two. When I say I’m gonna substitute it in, what it means is that I’m actually gonna put six plus 𝑦 instead of any of the 𝑥 values in equation two. And the reason we do this is so that we only have one variable. So in this case, it’s 𝑦. So now we can actually solve this equation for 𝑦 and find out the value or values of 𝑦.
Now the first stage is to expand the parentheses, because we have six plus 𝑦 all squared. What this means is six plus 𝑦 multiplied by six plus 𝑦. So first of all, we’re gonna have six multiplied by six, which is 36, then six multiplied by positive 𝑦, which gives us positive six 𝑦, then positive 𝑦 multiplied by six, which again gives us positive six 𝑦, and then finally 𝑦 multiplied by 𝑦, which gives us 𝑦 squared.
Okay, great. Now what we need to do is simplify this. So we can do this by collecting any like terms. So the two like terms are positive six 𝑦 and positive six 𝑦, cause they both have 𝑦 to the order of one. So therefore, we can say 36 plus 12𝑦 plus 𝑦 squared is the result of six plus 𝑦 multiplied by six plus 𝑦. Then we’ll have negative 54𝑦 — that’s because we’ve got negative nine 𝑦 multiplied by six; be careful here! there’s a common mistake to not include the negative sign; so do be careful when we have negatives — and then minus nine 𝑦 squared, because we have negative nine 𝑦 multiplied by 𝑦, which gives us negative nine 𝑦 squared. And then we have add 𝑦 squared equals 36.
So now that we’ve reached this stage, what we need to do is collect the like terms to simplify. So first of all, we have negative seven 𝑦 squared, and that’s because we’ve got positive 𝑦 squared minus nine 𝑦 squared, which will give us negative eight 𝑦 squared, plus 𝑦 squared, which will give us negative seven 𝑦 squared. Then we have minus 42𝑦. And this is because we have positive 12𝑦 minus 54𝑦, which gives us negative 42𝑦, then plus 36 equals 36.
So what we’re gonna do now is subtract 36 from each side of the equation, which is gonna leave us with negative seven 𝑦 squared minus 42𝑦 equals zero. So great, now what we can do is solve to find 𝑦. And to enable us to do that, we’re gonna factor. And if we factor negative seven 𝑦 squared minus 42𝑦, the first thing we’re gonna take outside the parentheses is negative seven 𝑦. And that’s because we have negative seven in each of our terms and we also have 𝑦 in each of our terms.
Then the first term inside the parentheses is gonna be 𝑦. That’s because negative seven 𝑦 multiplied by 𝑦 gives us negative seven 𝑦 squared. And then inside the parentheses for the second term, we’re gonna have positive six. And that’s because negative seven 𝑦 multiplied by positive six gives us negative 42𝑦.
Again, be careful of the negative here because often people will put negative inside the parentheses because they see the negative 42𝑦, but in fact a negative multiplied by a positive gives us a negative. So great, we’re now at the stage that we can solve and find out our values for 𝑦.
So therefore, 𝑦 equals zero or negative six. And the reason we got those two values is, first of all, we’ve got the zero value because we have to multiply negative seven 𝑦 by 𝑦 plus six to get the result of zero. Now if 𝑦 was equal to zero, we’d have zero on the outside of the parentheses multiplied by six on the inside. Well, zero multiplied by anything is zero. So that’s how we got our zero value. And we’ve got the negative six because, to get negative six, the inside of the parentheses would have to be equal to zero. So therefore, we set 𝑦 plus six equal to zero, subtract six from each side, and we get 𝑦 is equal to negative six.
So now we have our 𝑦 values, let’s move on and find our 𝑥 values. So in order to find our 𝑥 values, what we’re gonna do is substitute 𝑦 equals zero and 𝑦 equals negative six into equation three. So first of all, if you substitute in 𝑦 equals zero, we’re gonna get 𝑥 is equal to six plus zero. So we get a result of 𝑥 is equal to six. And if we substitute in 𝑦 equals negative six, we get 𝑥 is equal to six plus negative six. And if we add a negative, it’s the same as subtracting. So we have 𝑥 is equal to six minus six. So therefore, 𝑥 is gonna be equal zero.
So therefore, we can say the solutions to the simultaneous equations 𝑥 minus six equals six and 𝑥 squared minus nine 𝑥𝑦 plus 𝑦 squared equals 36 are 𝑥 equals six when 𝑦 equals zero and 𝑥 equals zero when 𝑦 equals negative six.