Solve: 1 over x-a= 1 over a-y and x+y over x-y =a

Question

This problem is an equation.

Answer 1

Are you asking for a solution for x and y in terms of a?  I am assuming that is what you seek.

 

(i)   1/(x - a) = 1/(a - y)
(ii)  (x + y)/(x - y)=a

 

From (i) get

 

x - a = a - y

x + y = 2a

 

From (ii) get

 

x + y = ax - ay

(1 - a)x + (1 +a)y = 0

 

Solve the system of linear equations in 2 unknowns x and y:

 

x + y = 2a

(1 - a)x + (1 +a)y = 0

 

Try substitution.  From top equation get

 

y = 2a - x

 

and plug into bottom equation to get x, plug the result for x into either equation to get y.

 

(1 - a)x + (1 + a)(2a - x) = 0

(1 - a)x + 2a - x + 2a² - ax = 0

(1 - a - 1 - a)x = -2a - 2a²

-2ax = -2a(1 + a)

 

x = 1 + a

 

 

y = 2a - x = 2a - (1 + a) = a - 1

 

y = a - 1

 

 

Test the solution.

 

 

 

 

Answer 2

equating DENOMINATORS of the first equation, we obtain   x-a = a-y  so x = 2a - y.

 

That might get you started; you can then substitute into the second equation, and (inshallah) solve for x.

Answer 3

first equation:

1/(x-a) = 1/(a-y) (take reciprocal of both sides)

x - a = a - y    (add a and add y to both sides)

x + y = 2a     (substitute this into second equation)

 

second equation:
(x+y)/(x-y)=a  (replace x+y in the numerator with 2a)
(2a)/(x-y) = a/1   ( cross multiply)

a(x-y) = 2a   (divide both sides by a & remember a≠0)

x - y = 2   (add y to both sides)

x = 2 + y

with restrictions: x≠y≠a and a≠0

(denominators would become zero)