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Intersecting Graphs


Date: 10/22/96 at 20:27:21
From: Geoff Peters
Subject: Intersecting Graphs

Hi Dr. Math!  

I'm an age 15 grade 10 student living in B.C. Canada, and I'm
wondering if you could help me with this "bonus" math problem.
(In this question the ^ denotes an exponent).
 
Intersecting graphs
 
The graph of x^2 + xy + y^2 = 3 intersects the graph of
x + xy + y + 1 = 0 in k distinct points.  What is the value of k?

Thanks so much for your time.  I'd really appreciate it if you could
give me an answer to that question! (My whole math class has not been 
able to get the correct answer.)


Date: 10/24/96 at 16:49:44
From: Doctor Rob
Subject: Re: Intersecting Graphs

The graphs of the two equations intersect at simultaneous solutions of 
the two equations.  First we will solve the second equation:

0 = x + xy + y + 1 = (x + 1)(y + 1)

This tells us that all solutions of this equation will have either 
x = -1 or else y = -1 (or both).  Now we solve each of these together 
with the first equation:

x = -1 and x^2 + xy + y^2 = 3 imply that 

    1 - y + y^2 = 3    or that

    y^2 - y - 2 = 0 = (y - 2)(y + 1)

so  y = -1 or y = 2.  

This gives two points, (x, y) = (-1, -1) and (-1, 2).

y = -1 and x^2 + xy + y^2 = 3 imply that 1 - x + x^2 = 3, or that
x^2 - x - 2 = 0 = (x - 2)(x + 1), so x = -1 or x = 2.  This gives two
points, (x, y) = (-1, -1) and (2, -1).

There are three solutions, namely (-1, -1), (-1, 2), and (2, -1).  
Thus k = 3.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Basic Algebra

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