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Question

solve for the letter
2by-2a=ay-4b :y
2x-a/b= 2x-b/a :x
x-a/ x-b= x-c/x-d : x
1/ay+1/by = 1/c :y

Answer 1

2by - 2a = ay - 4b
Collect the y terms on the left and the non-y terms on the right:
<=> 2by - ay = 2a - 4b
Now isolate y:
<=> (2b - a) y = 2a - 4b
<=> y = (2a - 4b) / (2b - a)

I assume the second one is
(2x-a)/b= (2x-b)/a
since otherwise we just get a/b = b/a <=> a = ± b.
Multiply through by ab:
<=> a(2x - a) = b(2x - b)
<=> 2ax - a^2 = 2bx - b^2
<=> 2ax - 2bx = a^2 - b^2
<=> 2 (a-b) x = (a-b) (a+b)
<=> x = (a+b) / 2 assuming a ≠ b - if a = b then the original equation is an identity.

(x - a) / (x - b) = (x - c) / (x - d)
Multiply through by (x - b) (x - d) to get
(x - a) (x - d) = (x - c) (x - b)
<=> x^2 - ax - dx + ad = x^2 - cx - bx + bc
<=> cx + bx - ax - dx = bc - ad
<=> (c + b - a - d) x = bc - ad
<=> x = (bc - ad) / (c + b - a - d), if c + b ≠ a + d.

1/ay + 1/by = 1/c: multiply through by abcy
<=> bc + ac = aby
<=> y = (bc + ac) / ab
= c/a + c/b.