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Advanced Math/x^3 - y^3 = 5803

Question
Dear Mr Salami,

I was wondering if you could help solve an argument.  The equation is x^3 - y^3 = 5803.  I got an answer of x = 20, y = 13 and x = -13, y = -20.  My friend says to solve this, just set y^3 = 0 and then the answer is y = 0, x = cube root 5803.  There seems to be something wrong with his answer, but I can't show mathematically if he's right or wrong and why.  Would you be able to help clarify things?

Sincerely,

Alex D

Hi Alex,
The first thing you need to realize is that the expression x^3 - y^3 = 5803 represents a family of points with specific values of x and y. Solving an equation by definition means to find particular values of the unknowns that satisfy the relationship. So we could solve, say
x² = 4
to get x = ±2 and these are the only numbers that satisfy the condition in the expression.
But when you have an expression involving more than one unknown, say
y = x²
then the value of one depends on the value of the other and so there is an infinite amount of pair values that satisfy this. It is so since we can allocate any value to x and y will subsequently get a value according to the statement. You can see this by plotting a graph to see that it represents a curve without bounds.
Back to your particular situation, your pair of x&y values would satisfy the expression, so would your friend's. Nothing is wrong with his answer since there's an infinite amount of x&y values that would be appropriate. Another of your friends could come and set y = 1 and still get a corresponding value for x, too.
In general, an expression that can be re-written in the form of y = f(x) which is read as "y is a function of x" would have values of y depending on x (which would be infinite in most cases). Now, there are times where we might require a certain set of values for the x&y pair. This would have to be accompanied by another condition. For example we know that there are an infinite amount of x&y points satisfying x^3 - y^3 = 5803. If we wanted the particular one where y is also equal to 2x, we would now have two expressions;
x^3 - y^3 = 5803
y = 2x
which could now be solved explicitly by substitution.
x^3 - (2x)^3 = 5803
x^3 - 8x^3 = 5803
-7x^3 = 5803
and result in a finite number of x&y values (only one real set) in this case.

I hope i have helped you.
You can always get back to me.

Regards

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Ahmed Salami

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I can provide good answers to questions dealing in almost all of mathematics especially from A`Level downwards. I can as well help a good deal in Physics with most emphasis directed towards mechanics.

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Aspiring theoretical physicist. I have been doing maths and physics all my life.

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I teach mathematics and engineering physics.