Substitution in fraction: Given that x/y = 2/7, evaluate (7x + y) / (x - (y/7))

[Dr.Peterson]

x/y is a fraction which is translated as 2/7. Fractions can also been seen as ratio. That is x/y = x:y= 2/7 = 2:7. The terms of the ratio are x and y or 2 and 7. That is why I am applying the second method.

But given only the ratio, x and y could be 4 and 14, or 6 and 21, and so on. They do not have to be 2 and 7.

Your method did work for this problem, but only because the expression (7x+y)/(x-(1/7)y) happens to be dependent only on the ratio, as shown by your first method. The second method assumes that, but does not prove it.

As I said in #3, you can overcome this by letting x=2k and y=7k; or, equivalently, you can do as @MarkFL said in #7, and replace 7x with 2y. These approaches show that the expression is independent of k, or of y alone.

What makes the problem work (so that there is a solution at all) is the the numerator and denominator are homogeneous. If it were something else, such as (7x+y)/(x-1), then there would be no solution -- but your second method would not reveal that.

Now, if you had said, "Because the numerator and denominator are homogeneous linear expressions [or something similar], we can use any specific terms of the ratio for x and y, ...", then it would have been valid (assuming you have a theorem to justify the claim).