Solve system by using identity...math????

Question

solve the system of xy = 1, x + y = a by using the identity:

((x -y) / 2)^2 = ((x + y) / 2)^2 - xy

??????? I have no idea ??????????

Answer 1

Do you remember the first two identities that we were taught in algebra classes?
These are
(x+y)^2=x^2+2xy+y^2 and (x-y)^2=x^2-2xy+y^2
we were also taught that
(x-y)^2=(x+y)^2-4xy
dividing both sides by 4,we get
(x-y)^2/4=(x+y)^2/4-4xy/4
or,{(x-y/2)}^2={(x+y/2}^2-xy
Now if you have the values of x+y and xy,input them in the above formula toget
{(x-y)/2}^2=(a/2)^2-1
or (x-y)/2}^2=a^2/4 -1=(a^2-4)/4
or,(x-y)/2= sqrt(a^2-4)/2
or,x-y=sqrt(a^2-4)
Now you have two equations
x+y=a and x-y =sqrt(a^2-4)
solving we get
x={a+sqrt(a^2-4)}/2
y={a-sqrt(a^2-4)}/2

Answer 2

Let us simplify the system first
((x-y)^2/2^2) = ((x+y)^2)/2^2) - xy
(x-y)(x-y)/4 = (x+y)(x+y)/4 -xy
multiply both sides with 4 we get
(x-y)(x-y)=(x+y)(x+y)-4xy
x^2-xy-xy+y^2=(x+y)(x+y)-4xy
Now use given information(xy=1,x+y=a) we get
x^2-1-1+y^2=(a)(a)-4
x^2 + y^2=a^2 - 2

Answer 3

Not difficult: 3x + 4y = -16, 3x + 11 = y... or; 3x - y = -11 Subtract out the second equation from the first. 5y = -5 y = -1 And now solve for x: 3x + 4y = -16, 3x + 11 = y 3x + (-4) = -16, 3x + 11 = (-1) 3x = -12, 3x = -12 x = -4, x = -4 y = -1 and x = -4 No mistaking it from the above. It's an ironclad proof!

Answer 4

yes i agree with the first answer, other wise ur just becoming a parrot in the sense that u dnt understand how to do it, but u just do them through memory... if u get that. i have no idea how 2 do that either lol, but i sugest u start off doing easier versions of this, then build up, practice makes perfect!!

Answer 5

((x -y) / 2)^2 = ((x + y) / 2)^2 - xy
((x -y) / 2)^2 = (a/2)^2 -1 =a^2/4-1 = (a2-4)/4
(x-y)/2 = sqrt(a^2-4)/2
x-y = sqrt(a^2-4)
x+y = a
2x = a+sqrt(a^2-4)
x = [a+sqrt(a^2-4)]/2
y = a- [a+sqrt(a^2-4)]/2

Answer 6

figure it out. Having someone else do it for you is not the answer. What are you learning if you dond't figure it out?

Answer 7

DO YOUR OWN HOMEWORK!!