Olympiad Mathematics | Simultaneously solved for you | Can You Solve This?

Added: 2026-05-24

[00:00:01]Okay, if you're ready, let's solve this equation simultaneously.

[00:00:08]Solution.

[00:00:11]Okay, we have x + y to be equal to five as our first equation.

[00:00:19]Then x * y equals five again is our second equation.

[00:00:29]Now, how do we solve this?

[00:00:31]We are going to use substitution method.

[00:00:35]So that from equation one, from our equation one, what do we do?

[00:00:44]We will say that x is equal to five minus y.

[00:00:49]From equation one, we've made x the subject.

[00:00:53]And this will be our equation three.

[00:00:57]Remember, we're going to need this equation three.

[00:01:00]Now, we're going to solve substitute into equation two.

[00:01:09]Now, if it is equation one that gave birth to equation three, put your equation three into equation two.

[00:01:17]And if it is equation two that gave birth to equation three, put your equation three into equation one.

[00:01:25]I hope that is okay.

[00:01:27]So our equation two is x y equals five.

[00:01:31]And we are now having that x is what?

[00:01:34]Five minus y.

[00:01:36]Then this y I have to write it. This is equal to five.

[00:01:42]Guess what I will do.

[00:01:45]Guess what I will do. I have to open the bracket. So that we have five y minus five squared equals five.

[00:01:55]Then, why don't we write the one with the highest power first?

[00:02:01]Right?

[00:02:02]As a matter of fact, I want to write y squared first.

[00:02:08]And y squared is going to be what?

[00:02:11]It's going to It's negative at the moment. Okay, you know what? Let's do it this way. So, we write our y squared.

[00:02:19]And then this is plus 5y.

[00:02:22]This is coming over to become minus five. This is equal to zero.

[00:02:27]I feel like explaining this again.

[00:02:31]Um wrote minus y squared first. This is plus 5y. This will turn to minus five.

[00:02:38]And it's all equal to zero.

[00:02:41]But we have to multiply all through by negative one. So, that the coefficient of y squared will be positive.

[00:02:49]So, this will make this positive. It will make this negative. It will make this positive.

[00:02:58]And zero is zero.

[00:03:02]Now, we have a quadratic equation. So, the question is how do we solve this quadratic equation? Is it what we can easily factorize?

[00:03:11]Or it is what we can use the quadratic formula. Now, to avoid thinking too much, use quadratic formula. Right? The formula now has a to be one. Coefficient of y squared, right?

[00:03:28]Then it has b to be minus five.

[00:03:31]Coefficient of y.

[00:03:33]And it has c to be the five, which is the constant.

[00:03:38]So, what is our formula?

[00:03:41]The formula is y being equal to minus b plus minus the square root of b squared minus 4ac as we divide all through by 2A.

[00:03:55]Don't stop there.

[00:03:57]Do not stop right there. Now, our Y is going to be putting the values of ABC.

[00:04:04]B is minus one. Look at that. Minus minus will turn to plus. So, we write positive.

[00:04:10]Plus or minus B squared is going to be minus five.

[00:04:15]And this is square on it.

[00:04:17]Then minus four times one times five.

[00:04:21]A is one. C is five. So, let's extend this.

[00:04:26]And we are dividing all of this by two multiplied by one.

[00:04:32]So, we will continue from here.

[00:04:36]So, we'll continue to get Y equals five plus minus then the square root of minus five squared is 25.

[00:04:46]Some persons will write minus 25, but that is wrong because it is minus five times minus five.

[00:04:54]Then minus four times five is 20.

[00:04:59]And we are dividing by two.

[00:05:02]So, we go on to get our Y to be five plus or minus the square root of five. That is five 25 minus 20.

[00:05:15]And we are dividing by two.

[00:05:17]So, what are we saying?

[00:05:20]We are saying that our Y is simply five plus the square root of five over two or five minus the square root of five over two.

[00:05:37]So, we will use the two values of X, I mean the two values of Y to get the corresponding values of Y of x.

[00:05:49]Now, remember from equation three that we said our x our x is equal to 5 minus y.

[00:05:59]So, in place of y, let's put the first value of y and our x will now be 5 minus we Okay, if you like, you open bracket and get your 5 plus the square root of 5 over 2.

[00:06:19]Right?

[00:06:20]And then we'll open the bracket so that x will be equal to 5 minus we have 5. You know what? This is going to confuse you. Yes, let me do it this way so that it will not confuse anybody.

[00:06:37]This is x to be equal to 5 minus you open the bracket. What we have here can be like 5 over 2 plus root 5 over 2.

[00:06:52]Okay, I believe it's better this way so that you can open the bracket so that your x will be 5 minus 5 over 2 minus the square root of 5 over 2.

[00:07:06]Do you know what I did?

[00:07:08]Negative is opening the bracket.

[00:07:11]Okay, so it's better this way. Now, your x is You find the LCM, which is 2. So, you're getting 2 over here.

[00:07:21]Then, 2 will multiply 5 to give you 10.

[00:07:25]Then here you're getting 5.

[00:07:28]Here you're getting your minus root 5.

[00:07:32]So, what do you do?

[00:07:34]You will now get your x to be equal to 10 minus 5 is 5.

[00:07:40]Then we have minus root five over two. So, this is the value of x.

[00:07:47]So, what are we saying?

[00:07:48]We are saying that when x is this y Okay, let me write on the other side.

[00:07:54]When x is this the value of y is equal to five plus the square root of five over two.

[00:08:05]Yes, this is it at this point. Now, we need to continue to get the other values.

[00:08:12]Right?

[00:08:14]Okay, so now our x we're going to get the second value second value of y which is five minus the square root of five over two.

[00:08:25]Now, if you like you do what I did before and if you don't you do what I'm about to do now, right?

[00:08:31]I want you to see the two ways.

[00:08:33]So, now our value of x is going to be x equals five the negative will multiply so that we can have negative five.

[00:08:43]The negative will turn that to multiplication. We have root five. But remember this is all over what?

[00:08:51]All over two.

[00:08:53]Okay, all over two. And this one here not all over two. It's from here to there that we have that.

[00:09:01]So, from here to here we divide by two just like we have here. But the negative will go in to change this to addition.

[00:09:09]Now, this is over one. We find the LCM like we did before. So, our x will now be two times five that is 10 then minus five plus root five.

[00:09:23]And all of this is over two.

[00:09:26]So, if we go on we will get x to be 10 minus five is five. Then we have plus root five.

[00:09:35]This is all over two.

[00:09:37]So, as I do points, the value of x is 5 + root 5 over 2. The value of y is 5 - root 5 over 2. Okay? So, by now we have solved the problem completely.

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