SOLUTION: solve 3x^3y^2-45x^2y=15x^3y-9x^2y^2 for x and y

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Question 1205914: solve 3x^3y^2-45x^2y=15x^3y-9x^2y^2 for x and y
Found 3 solutions by MathLover1, greenestamps, ikleyn:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

3x%5E3y%5E2-45x%5E2y=15x%5E3y-9x%5E2y%5E2.....factor
3+x%5E2+%2Ay+%28x+y+-+15%29+=+3+x%5E2%2A+y+%285+x+-+3+y%29
3+x%5E2+%2Ay+%28x+y+-+15%29+-+3+x%5E2+%2Ay+%285+x+-+3+y%29=0
3+x%5E2+%2Ay+%28%28x+y+-+15%29+-++%285+x+-+3+y%29%29=0
3+x%5E2+%2Ay+%28x+y+-+15+-++5+x+%2B3+y%29=0
3+x%5E2+%2Ay+%28%28x+y%2B3+y%29+-+%285x+%2B+15%29+%29=0
3+x%5E2+%2Ay+%28y%28x+%2B3+%29+-+5%28x+%2B+3%29+%29=0
3+x%5E2+%2Ay+%28y+-+5%29%28x+%2B+3%29+=0



solutions:
if 3+x%5E2+%2Ay++=0+=> x=0 (periodicity 2), y=0
if+%28y+-+5%29+=0=> y=5
if+%28x+%2B+3%29+=0=> x=-3

answer:
x=0
x=-3
y=0
y=5

Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


3x%5E3y%5E2-45x%5E2y=15x%5E3y-9x%5E2y%5E2

Move everything to one side with the other side zero.

3x%5E3y%5E2-45x%5E2y-15x%5E3y%2B9x%5E2y%5E2=0

Factor the common factor out of all the terms.

%283x%5E2y%29%28xy-15-5x%2B3y%29=0

Group the terms in the second factor and look for a common factor.

%283x%5E2y%29%28%28xy-5x%29%2B%283y-15%29%29=0
%283x%5E2y%29%28x%28y-5%29%2B3%28y-5%29%29=0
%283x%5E2y%29%28x%2B3%29%28y-5%29=0

Use the basic idea that the product is zero only if at least one of the factors is zero:

ANSWERS: x=0; y=0; x=-3; y=5

Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.

In order for you better understand the answer, I will give an expanded answer. 

    There are four infinite sets of solutions:

          (a)  {x= 0;  y is any real number};

          (b)  {x= -3; y is any real number};

          (c)  {y= 0;  x is any real number};

          (d)  {y= 5;  x is any real number}.