✔ 最佳答案
The answer from the above respondent only tells you (x + y) varies directly as (x² + y²)/(x - y). If we want to prove (x+y) varies directly as (x-y), we have to remove all variables except (x+y) and (x-y).
If (x²+y²) varies directly as (x²-y²), we have
(x² + y²) = k(x² - y²), where k is a non-zero constant.
y² + ky² = kx² - x²
y² (1 + k) = x² (k - 1)
Thus x² = (1+k)/(k-1) X y²
Note that (1+k)/(k-1) is in fact another constant, so let c² = (1+k)/(k-1).
We have x² = c²y²
Taking square roots on both sides, x = cy. In other words, x varies directly as y.
Consider (x+y) / (x-y)
(x+y) / (x-y)
= (cy + y) / (cy - y)
= [y(c+1)] / [y(c-1)]
= (c+1) / (c-1)
Note that c is a constant, (c+1) / (c-1) is also a constant.
That is, (x+y) / (x-y) = constant.
Therefore (x+y) varies directly as (x-y).