x+y = 4z, x*y = 4z^2, express x-y in z

To solve the equation x + y = 4 z and x * y = 4 z ^ 2 , we can use algebra ic manip ulation . \n First , we can sub tract x * y from both sides of the first equation to get : \n x + y - x * y = 4 z - x * y \n Sim pl ifying this equation , we get : \n y = 4 z - x * y \n Next , we can substitute this expression for y into the second equation to get : \n x * ( 4 z - x * y ) = 4 z ^ 2 \n Exp and ing this equation , we get : \n 4 x z - x ^ 2 y = 4 z ^ 2 \n R ear r anging this equation , we get : \n x ^ 2 y - 4 x z + 4 z ^ 2 = 0 \n This is a quadr atic equation in terms of x and y . We can solve it using the quadr atic formula : \n x = (- b ± sq rt ( b ^ 2 - 4 ac )) / 2 a \n where a = 1 , b = - 4 , and c = 4 z ^ 2 . \n Sub st itut ing these values , we get : \n x = ( 4 ± sq rt ( 1 6 - 1 6 z ^ 2 )) / 2 \n Sim pl ifying this equation , we get : \n x = ( 1 ± sq rt ( 1 - z ^ 2 )) / 2 \n Now we can substitute this expression for x into the equation y = 4 z - x * y to solve for y . \n y = 4 z - ( 1 + sq rt ( 1 - z ^ 2 )) / 2 * y \n Mult ip lying both sides by 2 y , we get : \n 2 y = 4 z - ( 1 + sq rt ( 1 - z ^ 2 )) \n Add ing ( 1 + sq rt ( 1 - z ^ 2 )) to both sides , we get : \n 3 y = 4 z \n D ivid ing both sides by 3 , we get : \n y = 4 z / 3 \n There fore , the solution to the equation x + y = 4 z and x * y = 4 z ^ 2 is x = ( 1 + sq rt ( 1 - z ^ 2 )) / 2 and y = 4 z / 3 . </s>