SOLUTION: How many solutions does the following non-linear system of equations have? x2 - 2y = 8 x2 + y2 = 16 I would really appreciate some help thank you

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Question 254520: How many solutions does the following non-linear system of equations have?
x2 - 2y = 8
x2 + y2 = 16
I would really appreciate some help thank you
Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

Actually, the short answer is 4. We know that because when you solve the first equation for in terms of , you will have an expression in . When this expression is substituted into the second equation then you will have an expression in which will lead to a quartic equation which solution set are the -coordinates of the ordered pairs that comprise the solution set of the system. The Fundamental Theorem of Algebra says that any -th degree polynomial equation has exactly roots. Hence, the 4th degree equation will have 4 roots and the system of equations will have 4 solutions.

Let's verify:

Solve for

(verification of the last step is left as an exercise for the student)

Next, substitute this expression into the second equation:

Expand the binomial and simplify:

(again, verification of the algebra is left as an exercise for the student)

Factor the LHS:

Hence

or

If

then

Root 1

or

Root 2

If

then

Root 3

or

Root 4

Now all that is necessary is to substitute the 4 values obtained for into either equation (the first one gives us simpler arithmetic) and solve for to develop the 4 ordered pairs that comprise the solution set of the original system.

so the first ordered pair is:

so the second ordered pair is:

so the third ordered pair is:

so the 4th ordered pair is:

But how can there be 4 solutions when 2 of them are identical? All I can say is refer back to the Fundamental Theorem of Algebra. Albert Girard said it, Jean-Robert Argand proved it, I believe it, that settles it.

John