Question 1101236
 make x the subject of the formula 
 1. {{{x/(x+a) - a/(x+b) = 1}}}
multiply by (x+a)(x+b), cancel the denominators
x(x+b) - a(x+a) = (x+a)(x+b)
distribute, FOIL
x^2 + bx - ax - a^2 = x^2 + bx + ax + ab
combine like terms
x^2 - x^2 - ax - ax = a^2 + bx - bx + ab
-2ax =  a^2 + ab
{{{x = (a^2+ab)/(-2a)}}}
factor out a
{{{x = (a(a+b))/(-2a)}}}
Cancel a
{{{x = -(a+b)/2}}}
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 2. {{{x^2/a^2 + y^2/b^2 = 1}}}
multiply equation by a^2b^2
{{{b^2x^2 + a^2y^2 = a^2b^2}}}
{{{b^2x^2 = a^2b^2 - a^2y^2}}}
{{{b^2x^2 = a^2(b^2-y^2)}}}
{{{x^2 = (a^2(b^2-y^2))/b^2}}}
{{{x = sqrt((a^2(b^2-y^2))/b^2)}}}
extract the squares
{{{x = (a/b)sqrt(b^2-y^2)}}}
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Will finish later today, these are time consuming.
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 3. {{{a(a^2- x) = b(b^2 - x)}}}
a^3 - ax = b^3 - bx
bx - ax = b^3 - a^3
x(b-a) = b^3 - a^3
x = {{{(b^3-a^3)/(b-a)}}}
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 4. {{{a/(a-x) = b/(b+x)}}} 
cross multiply
a(b+x) = b(a-x)
ab + ax = ab - bx
bx + ax = ab - ab
bx + ax = 0
x(b + a) = 0
x = {{{0/((b+a))}}}
x = 0