Expert: Paul Klarreich - 5/14/2007

Question
Hello,

I'll try to meet all the criteria of your instructions, but since this question isn't mine, I might not be able to meet them all.  :)  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
Questioner:   Yells
Category:  Advanced Math
 
Subject:  Algebra/Calculus???
Question:  Hello,

I'll try to meet all the criteria of your instructions, but since this question isn't mine, I might not be able to meet them all.  :)  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

>> Yes, I agree.  The 'AND' is the missing word that allows the example to make some sense.

What is the value of XY?  Which will give a simple answer of -5.  Easy.

>> Hmmm.  I wonder what you meant by that.

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:
¡î5 (Y + 15)^2 + ¡îY^2 = ¡î15

>> OH , NO!  That is definitely bad algebra.  You can't just take the square root of each term.  NO NO NO NO NO!



5 (Y + 15) + Y = ¡î15
6Y©÷ + 75©÷ = ¡î15©÷
6Y©÷ + 5625 = 15
-5625

¡î6Y©÷ = ¡î-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!
.............................
Hi, Yells,

On the assumption that the question really says:
x = y + 5   AND  x^2 + y^2 = 15

Then we have a system of simultaneous equations in two variables.  It is not linear (has squares) so we have to use substitution to solve it:

x - y = 5   gives  x = y + 5

which we substitute into:

x^2 + y^2 = 15

(y + 5)^2 + y^2 = 15
y^2 + 10y + 25 + y^2 = 15

2y^2 + 10y + 10 = 0

y^2 + 5y + 5 = 0

You will have to use the quadratic formula.  The solutions (YES, PLURAL) are:

   - 5 +- sqrt(25 - 4(1)(5))
y = -------------------------
              2

   - 5 +- sqrt(25 - 20)
y = -------------------------
              2

   - 5 +- sqrt(5)
y = ---------------
          2

Now go back and get the solutions for x.  Use the same substitutor:

x = y + 5

   - 5 +- sqrt(5)
x = --------------- + 5
          2

   - 5 +- sqrt(5) + 10
x = -------------------
          2
   5 +- sqrt(5)
x = ------------
       2

You did want the value of xy, didn't you?  Multiply those:

         5 +- sqrt(5)   -5 +- sqrt(5)
Now  xy = -------------- ------------- =
              2               2

The numerators are conjugates (identical except for one sign) and that make the multiplication come out nicely.

    -25 + 5    -20
xy = -------- = ---- = - 5
       4         4

Well, there's your -5.  In answer to your other question: Since it involves the quadratic formula and simultaneous equations, this is Intermediate Algebra, normally taught in the 11th grade in the U.S.
....................................
Now if you have to solve THIS set of equations (which does not make much sense) it would go like this:

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

y = x - 15

5x^2 + (x - 15)^2 = 15

5x^2 + x^2 - 30x + 225 = 15

6x^2 - 30x + 210 = 0

x^2 - 5x + 35 = 0
   5 +- sqrt(25 - 140)
x = -------------------
          2
   5 +- sqrt(- 115)
x = -----------------
          2
   5 +- i sqrt(115)
x = -----------------
          2
etc.