How to solve the following equations

(1)a²  + b² =  x and a + b = y 

(2) a²  + b² =  x and a - b = y

(3) a²  - b² =  x and a + b = y and

(4) a²  - b² =  x and a - b = y

If a+b = x and a-b = y then a = (x+y)/2 and b = (x-y)/2

(1) a²  + b² =  x and a + b = y ; (a+b)² = y² = a²  + b² + 2ab = x + 2ab ; y² – x = 2ab or b = (y² – x)/2a

      a + b = y; a + (y² – x)/2a = y or 2a²  - 2ay + (y² – x) = 0 ; a = (y/2) + ½ √ (2x –y² ) and b = (y/2) - ½ √ (2x –y² )

(2) a²  + b² =  x and a - b = y; (a-b)² = y² = a²  + b² - 2ab = x - 2ab ; x – y² = 2ab or b = (x- y² ) /2a

       a - b = y; a - ( x - y²)/2a = y or 2a²  - 2ay - (x - y²) = 0 ;    a = (y/2) + ½ √ (2x –y² ) and b =- (y/2) + ½ √ (2x –y² )

(3) a²  - b² =  x and a + b = y ;  a²  - b² = (a+b)(a-b) = y (a-b) =  x or a-b = x/y ; a+b= y

      a = (y² +x)/2y and b = (y² - x)/2y

(4) a²  - b² =  x and a - b = y; a²  - b² = (a+b)(a-b) = y (a+b) = x or a + b = x/y ; a-b = y

      a = (x + y² )/2y and b = (x - y²)/2y