If two distinct non-zero real variables 𝑥 and 𝑦 are such that (𝑥 + 𝑦) is…

Let x+y be proportional to x−y, so there exists a constant k such that x+y = k(x−y).

Rearrange and collect like terms:

• x + y = kx − ky

• Bring x-terms to one side: x(1 − k) = −y(1 + k)

Divide both sides by y(1 − k) (noting 1 − k ≠ 0):

x/y = −(1 + k)/(1 − k)

Thus for the given proportionality constant k the ratio x/y is a fixed number, so x/y is a constant. Note k cannot equal 1 because that would make the denominator zero; k = −1 would give x = 0, which is excluded since x is non-zero.

Example: if k = 2 then x/y = −(1+2)/(1−2) = −3/−1 = 3, illustrating that once the proportionality constant is fixed the ratio is fixed.