SOLUTION: i have some problems with this system ? can you tell me how to solution sytem sytem is x^3 + y^3 = 1 x^2y + 2xy^2 + Y^3 = 2 please contact with me as soon as you ca

Algebra ->  Systems-of-equations -> SOLUTION: i have some problems with this system ? can you tell me how to solution sytem sytem is x^3 + y^3 = 1 x^2y + 2xy^2 + Y^3 = 2 please contact with me as soon as you ca      Log On


   



Question 167734: i have some problems with this system ? can you tell me how to solution sytem
sytem is x^3 + y^3 = 1
x^2y + 2xy^2 + Y^3 = 2
please contact with me as soon as you can

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
system%28x%5E3%2By%5E3=1%2C+x%5E2y%2B2xy%5E2%2By%5E3=2%29

Let y=kx









Now we can solve each for x%5E3 as long as we require that
1%2Bk%5E3%3C%3E0 and k%2B2k%5E2%2Bk%5E3%3C%3E0

So we will take Case 1 as when they are both true:

Case 1:  1%2Bk%5E3%3C%3E0 and k%2B2k%5E2%2Bk%5E3%3C%3E0

Then we can solve both for x%5E3 without
dividing by 0:

system%28x%5E3=1%2F%281%2Bk%5E3%29%2C+x%5E3=2%2F%28k%2B2k%5E2%2Bk%5E3%29%29

Since the left sides are equal, so are the right 
sides:

1%2F%281%2Bk%5E3%29=2%2F%28k%2B2k%5E2%2Bk%5E3%29

Cross multiply:

2%281%2Bk%5E3%29=1%28k%2B2k%5E2%2Bk%5E3%29

2%2B2k%5E3=k%2B2k%5E2%2Bk%5E3

Get 0 on the right and left side in descending
order:

k%5E3-2k%5E2-k%2B2=0

Factor k^2 out of the first two terms on the left,
and factor -1 out of the last two terms on the left:

k%5E2%28k-2%29-1%28k-2%29=0

Factor k-2%29 out of both terms on the left:

%28k-2%29%28k%5E2-1%29=0

Factor the second parentheses on the left:

%28k-2%29%28k-1%29%28k%2B1%29=0

Using the zero-factor principle:

 

Remember that the requirements for Case 1 were:

1%2Bk%5E3%3C%3E0 and k%2B2k%5E2%2Bk%5E3%3C%3E0

We investigate to see whether k=2 satisfies
both of those requirements. Substituting,

1%2Bk%5E3%3C%3E0 and k%2B2k%5E2%2Bk%5E3%3C%3E0
1%2B2%5E3%3C%3E0 and 2%2B2%282%29%5E2%2B%282%29%5E3%3C%3E0
1%2B8%3C%3E0 and 2%2B2%284%29%2B8%3C%3E0
9%3C%3E0 and 2%2B8%2B8%3C%3E0
9%3C%3E0 and 18%3C%3E0

So there are both satisfied, so we can use k=2

We now investigate to see whether k=1 satisfies
both of those requirements. Substituting,

1%2Bk%5E3%3C%3E0 and k%2B2k%5E2%2Bk%5E3%3C%3E0
1%2B1%5E3%3C%3E0 and 2%2B2%281%29%5E2%2B%281%29%5E3%3C%3E0
1%2B1%3C%3E0 and 2%2B2%281%29%2B1%3C%3E0
2%3C%3E0 and 2%2B2%2B1%3C%3E0
2%3C%3E0 and 5%3C%3E0

So these are both satisfied, so we can also use k=1

We now investigate to see whether k=-1 satisfies
both of those requirements. Substituting,

1%2Bk%5E3%3C%3E0 and k%2B2k%5E2%2Bk%5E3%3C%3E0
1%2B%28-1%29%5E3%3C%3E0 and 2%2B2%28-1%29%5E2%2B%28-1%29%5E3%3C%3E0
1-1%3C%3E0 and 2%2B2%281%29-1%3C%3E0
0%3C%3E0 and 2%2B2-1%3C%3E0
0%3C%3E0 and 3%3C%3E0

Since the first is not satisfied we cannot use k=-1

Now we usek=1 and since y=kx, y=x in the original:

system%28x%5E3%2By%5E3=1%2C+x%5E2y%2B2xy%5E2%2By%5E3=2%29
system%28x%5E3%2Bx%5E3=1%2C+x%5E2x%2B2xx%5E2%2Bx%5E3=2%29
system%282x%5E3=1%2C+x%5E3%2B2x%5E3%2Bx%5E3=2%29
system%282x%5E3=1%2C+4x%5E3=2%29

solving both for x%5E3

system%28x%5E3=1%2F2%2Cx%5E3=1%2F2%29

So these are the same, so we can

get the real solution by taking cube roots
of both sides:

x=root%283%2C1%2F2%29

rationalizing:

x=root%283%2C%281%2A4%29%2F%282%2A4%29%29+

x=root%283%2C4%2F8%29

x=root%283%2C4%29%2F2%29

And since in this case y=x

we have solution

 

But that's not necessarily the only solution.

Now we usek=2 and since y=kx, y=2x in the original:

system%28x%5E3%2By%5E3=1%2C+x%5E2y%2B2xy%5E2%2By%5E3=2%29
system%28x%5E3%2B%282x%29%5E3=1%2C+x%5E2%282x%29%2B2x%282x%29%5E2%2B%282x%29%5E3=2%29
system%28x%5E3%2B8x%5E3=1%2C+2x%5E3%2B2x%284x%5E2%29%2B8x%5E3=2%29
system%289x%5E3=1%2C+2x%5E3%2B8x%5E3%2B8x%5E3=2%29
system%289x%5E3=1%2C+18x%5E3=2%29
system%289x%5E3=1%2C+9x%5E3=1%29

solving both for x%5E3

system%28x%5E3=1%2F9%2Cx%5E3=1%2F9%29

So these are the same, so we can
get a real solution by taking cube roots
of both sides:

x=root%283%2C1%2F9%29

rationalizing:

x=root%283%2C%281%2A3%29%2F%289%2A3%29%29+

x=root%283%2C3%2F27%29

x=root%283%2C3%29%2F3%29

And since in this case y=2x

we have solution

 

So we have found two solutions.  But this
was only for Case 1, when 

1%2Bk%5E3%3C%3E0 and k%2B2k%5E2%2Bk%5E3%3C%3E0

We must investigate the possibility of
solutions when one or both of these are
violated:

Case 2: 1%2Bk%5E3=0

This means that

k%5E3=-1 or k=-1

Now we usek=-1 and since y=kx, y=-x in the original:

system%28x%5E3%2By%5E3=1%2C+x%5E2y%2B2xy%5E2%2By%5E3=2%29
system%28x%5E3%2B%28-x%29%5E3=1%2C+x%5E2%28-x%29%2B2x%28-x%29%5E2%2B%28-x%29%5E3=2%29
system%28x%5E3-x%5E3=1%2C+-x%5E3%2B2x%5E3-x%5E3=2%29
system%280=1%2C+0=2%29

This is certainly not true, so Case 2
1%2Bk%5E3=0 is not possible since it produces
no solutions.

Case 3: k%2B2k%5E2%2Bk%5E3=0

Factoring out k on the left:

k%281%2B2k%2Bk%5E2%29=0

Factoring the trinomial in parenheses:

k%281%2Bk%29%281%2Bk%29=0

Using the zero-factor principle:



We have already seen that k=-1 is not
possible, so we only need investigate k=0

Now we usek=0 and since y=0x, y=0 in the original:

system%28x%5E3%2By%5E3=1%2C+x%5E2y%2B2xy%5E2%2By%5E3=2%29
system%28x%5E3%2B%280%29%5E3=1%2C+x%5E2%280%29%2B2x%280%29%5E2%2B%280%29%5E3=2%29
system%28x%5E3=1%2C+0=2%29
system%28x=1%2C+0=2%29

This is certainly not true, so Case 3
k=0 is not possible since it produces
no solutions.

So the only real solutions are:



and

 

However, there are also some imaginary solutions.  I did not
find those. They would require finding the 2 imaginary
cube roots each of 4 and 3.  I only considered the real
cube roots of 4 and 3.  Post again if you want all the
imaginary solutions as well.

Edwin