Question

2x+4=lambda(2x)
2y-4=lambda(2y)
(x^2)+(y^2)=1

Solve for x and y

Answer 1

Yay for La Grange multipliers! I just took this test this morning....
So if we isolate x and y in terms of lambda, we can solve for lambda in the third equation and then find x and y using the first two equations again...
So the first equation becomes:
x( 1- lambda) + 2 = 0 -> x= -2/(1- lambda)
The second equation becomes:
y ( 1- lambda) -2 = 0 -> y= 2/(1 - lambda)
Then if we use these x and y values in the third equation we get:
4/(1- lambda)^2 + 4/(1- lambda)^2 = 1
Solve for lambda and we get :
lambda= 1- sqrt(8)
Since we now know lambda, we can sub that value back into the equations where we isolated x and y in terms of lambda, and we get:
x= -(sqrt2)/2
y= (sqrt 2)/2