Can anyone answer this?

Question

the first question on an examination paper is:

Solve for x the equation: 1/x = 1/a + 1/b

where (in the question) a and b are given non-zero real numbers. One candidate writes x = a + b as the solution. Show that there are no values of a and b for which this will give the correct answer.

Answer 1

Here is a simple proof of this problem.

1/x = 1/a + 1/b = (a+b) / ab ---> x = ab / (a + b).

One candidate asserts that x = a + b. To test his assertion, all we need to do is substitute his claimed value for x in our equation. If we do that, then we get:

x = (a + b) = ab / (a + b).

The above equation implies the following:

x = (a + b)² = a² + 2ab + b² = ab.

We see that the last equation is an impossibility unless a = b = 0, in which case we would violate the provision that a and b are non-zero real numbers. So this candidate's assertion can be totally rejected, and he can be sent back to the hinterland for further education and edification.

Answer 2

1/x=1/a+1/b
take LCM
1/x=b+a/ab
take resiprocal
x=ab/(a+b)

Answer 3

1/x = 1/a + 1/b

x = ab/(a+b)

Answer 4

1/x=1/a+1/b

x=1/(1/a+1/b)
x=1/(b/(a*b)+a/(a*b))
x=1/((a+b)/(a*b))
x=a*b/(a+b)

Answer 5

x=ab/(a+b)