Question

puteti sa rezolvati urgent g),h),i)​

Answer 1

a)

1/x-1/y = 0

2/x+3/y-5 = 0

1/x = a, 1/y = b

a-b = 0     |*3

2a+3b-5 = 0

3a-3b=0

2a+3b-5 = 0

5a-5 = 0 => 5a = 5 => a = 1 => b = 1 => 1/x = 1 => x = 1 => 1/y = 1 => y = 1

b)

(x+y)/xy = 3

(x-y)/xy = -1 => (x^2-2xy+y^2)/(xy)^2 = 1 => ((x+y)^2-2xy-2xy)/(xy)^2 = 1 =>

=> ((x+y)^2-4xy)/(xy)^2 = 1

x+y = a

xy = b

=> a/b = 3

=> (a^2-4b)/b^2 = 1 => (a/b)^2 - 4/b = 1 => 3^2-4/b = 1 => 9-4/b = 1 => 4/b = 8 => b = 1/2

=> 2a = 3 => a = 3/2

=> x+y = 3/2

=> xy = 1/2

=> t^2-(3/2)t+1/2 = 0 => 2t^2-3t+1 = 0

Delta = 9-8 = 1 => t1,2 = (3±1)/4 => t1 =1/2 => t2 = 1

=> x = 1/2, y = 1

c)

2/(x+2y)-3/(x-y) = 5

1/(x+2y)+1/(x-y) = 0

x+2y = a, x-y = b

2/a - 3/b = 5

1/a + 1/b = 0 |*3

2/a - 3/b = 5

3/a + 3/b = 0

5/a = 5 => a = 1 => 2-3/b = 5 => -3/b = 3 => b = -1

=> x+2y = 1

=> x-y = -1 |*2

=> x+2y = 1

=> 2x-2y = -2

=> 3x = -1 => x = -1/3

=> -1/3+2y = 1 => 2y = 4/3 => y = 2/3