Expert: Steve Holleran - 5/14/2007

Question
Hello,

I saw this question on a message board:

X-Y=5 X^+2  Y^2 = 15
What is the value of XY?

It occurs to me now that they meant "X-Y=5 and X^2 + Y^2 = 15 What is the value of XY?"  Which will give a simple answer of -5.  Easy.

HOWEVER, I misunderstood it to mean:

X-Y = 5X^2 + Y^2 = 15
What is the value of XY?

(Also written as:)

X-Y = 15
5X^2 + Y^2 = 15

I solved this and got: 185.7i - 156.25 = XY
My work on the problem is as follows:
√5 (Y + 15)^2 + √Y^2 = √15

5 (Y + 15) + Y = √15
6Y² + 75² = √15²
6Y² + 5625 = 15
-5625

√6Y² = √-5610
6Y = 74.9i
/6
Y = 12.5i

Now replace every Y with 12.5i
X -12.5i = 15
+ 12.5i
X = 15 + 12.5i
(15 + 12.5i)(12.5i)=
187.5i + 156.25i^2
XY = 185.7i - 156.25

I do not know if this was correct because I have not gotten any further than precalculus and statistics in school.  Can you please tell me:

A) What subject is this from?  (i.e. Statistics, Calculus, etc.)
B) Was my answer correct?  If not, can you please tell me what the answer would be and how you arrived at that answer?

Thank you so much!

Answer
Hello Tree,

Well, your interpretation which gave you -5 as the solution is the correct one.  Its basically a "systems of equations" problem which could come up in some Algebra 1 classes, but more likely would be seen in Algebra 2.

On the steps you outlined, I believe there is one major error early on which renders the rest unacceptable.  Let me see if I understand what you did:

From      x - y = 15

and      5x^2  + y^2 = 15   I can see you solved for x and substituted to get the line:

        5(y + 15)^2 + y^2 = 15.

Its the next step which is "illegal":

You can't take the square root of each term on the left separately; you have to take the square root of the entire side.  It should be:

     sqrt[5(y + 15)^2 + y^2] = sqrt 15

NOT   sqrt[5(y + 15)^2] + sqrt[y^2] = sqrt 15,

which changes the problem altogether.

If you had to solve it the way you asked about, I would do the following:

              5(y + 15)^2 + y^2 = 15

        5(y^2 + 30y + 225) + y^2 = 15

        5y^2 + 150y + 1125 + y^2 = 15

        6y^2 + 150 y + 1110 = 0

or         y^2 + 25y + 185 = 0

using the quadratic formula, you would get imaginary roots of -12.5 +/- 5.36i

Hope this is what you were looking for.

Steve Holleran