A pair of equations.

Algebra Level 4

How many ordered pairs of real numbers ( x , y ) (x,y) satisfy: ( x − 1 ) ( y 2 + 6 ) = y ( x 2 + 1 ) , (x-1)(y^{2}+6)= y(x^{2}+1), ( y − 1 ) ( x 2 + 6 ) = x ( y 2 + 1 ) ? (y-1)(x^{2}+6)=x(y^{2}+1)?


The answer is 4.

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Abhishek Singh by Abhishek Singh , 24, India

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3 solutions

Jack D'Aurizio
Jun 29, 2014

If you sum the two equations you get the equation of a circle, while if you subtract them you get ( x − y ) (x-y) times the equation of an hyperbola. Simple algebra follows.

5 Helpful 0 Interesting 0 Brilliant 0 Confused
Saud Molaib
Sep 12, 2014

Notice that the two relations are inverses of each other since each y in the first relation is interchanged for an x in the second and each x in the first relation is interchanged for a y in the second equation. Inverse relations only intersect on the line of symmetry y=x.

This means we can substitute each y for an x in one of the equations. When we simplify we get a parabola with a positive discriminant in the variable x giving us two solutions. The same can be done if we replace each x for a y. The process is identical to the last and gives us two more different roots. Thus we have 4 solutions to the system of equations.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Could you explain a little bit your method?

Juan rodrígez - 6 years, 9 months ago

Note that ( 2 , 3 ) (2,3) is a solution to the problem, which contradicts your claim.

Using your approach, the only solutions to

y = − x y = -x x = − y x = -y

would be when x = y x = y , hence x = y = 0 x = y = 0 . But we know that there are many other solutions with x ≠ y x \neq y .

Calvin Lin Staff - 6 years, 8 months ago

cyclic order (x-1)(x^2 + 6 ) = x(x^2 + 1 ) solve x = 2 , 3 x=2 in equation1 solve y = 2 , 3 x=3 in equation1 solve y = 2 , 3 ( x , y ) = ( 2 , 2 ) , ( 2 , 3 ) ( 3 , 2 ) , ( 3 , 3 ) Ans = 4 pairs.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

You should explain what you meant by "cyclic order (x-1)(x^2 + 6 ) = x(x^2 + 1 )". It looks unclear.

mathh mathh - 6 years, 11 months ago

As I understand, cyclic order means replace x with y AND y with x. You have only replaced y with x. Please explain the theory behind your method.

Niranjan Khanderia - 6 years, 11 months ago

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