# Hard simultaneous equations

Hi

How do you solve this equations analyticaly without solving a 4th degree polynomial
sqrt(x)+y=11; x+sqrt(y)=7

Thanks

I'll give a general outline on how to solve this:

First of all, use new variables. a = sqrt(x) and b = sqrt(y). That will remove those ugly square roots for now.

Then, add the 2 equations and form a new equation a + a² + b + b² = 18. In this equation, 18 can be split up into 2 unknown values c and d to form 2 quadratic equations. Of these 2 values, you only need to know 1 to calculate the other, that is d = 18 - c. The roots of these 2 quadratic equations can be written in terms of these 2 values. Keeping in mind that the solutions can only be positive, that should lead you to the right answer

Last edited:
HallsofIvy
Homework Helper
I may be misunderstanding but you seem to be saying that if you have $a+ a^2+ b+ b^2= 18= c+ d$ then you must have $a+ a^2= c$ and $b+ b^2= d$. That is not true.

Last edited by a moderator:
I may be misunderstanding but you seem to be saying that is you have $a+ a^2+ b+ b^2= 18= c+ d$ then you must have $a+ a^2= c$ and $b+ b^2= d$. That is not true.
Do u want to explain why this is not true? If I define c as being a+a², and d as being b+b², then a + a² + b + b² = c + d = 18

Edit: nvm, you probably did misunderstand me^^

I'll give a general outline on how to solve this:

First of all, use new variables. a = sqrt(x) and b = sqrt(y). That will remove those ugly square roots for now.

Then, add the 2 equations and form a new equation a + a² + b + b² = 18. In this equation, 18 can be split up into 2 unknown values c and d to form 2 quadratic equations. Of these 2 values, you only need to know 1 to calculate the other, that is d = 18 - c. The roots of these 2 quadratic equations can be written in terms of these 2 values. Keeping in mind that the solutions can only be positive, that should lead you to the right answer
can you plees put full steps for this equation plees

Hello I'm 14 years old so i don't know what a 4th degree polynomial is... however i agree with the guy named Kyouran. This is how i did it:

sqrt(x) + y = 11 ...(1)
sqrt(y) + x = 7 ...(2)

Let sqrt(x) = a, and sqrt(y) = b.

The eqs will then be:
a + b^2 = 11 ...(3)
b + a^2 = 7 ...(4)

Add them together and you get:

a + a^2 + b + b^2 = 18

The way i solved it is just by doing guess and check. This only takes common sense:

I first tried a = 2 and b = 3 which when substituted into the equation equals 18:

2 + 2^2 + 3 + 3^2 = 18

However when subbing them into eqs 3 and 4, they weren't correct:

So i then switched the values around saying a = 3, and b = 2 which when subbed into eqs 3 and 4 were correct!

And so, as said before...

Since I let sqrt(x) = a, and the sqrt(y) = b:

sqrt (x) = 3
and so therefore x = 9

sqrt (y) = 2
and so therefore y = 4

therefore x = 9, y = 4 ;)

Now you will find, that when subbing these x and y values into your original eqs (1) and (2), they are sure to be correct!

Last edited: