Question

If 2/x+ 3/y =13 and 5/x- 4/y = -2, then x+y equals

Answer 1

Given that

(2/x)+(3/y) = 13--------(1)

(5/x)-(4/y) = -2 -------(2)

Put 1/x = a and 1/y = b then

2a + 3b = 13 ----------(3)

On multiplying with 5 then

10a +15 b = 65 -------(4)

and

5a -4b= -2 ----------(5)

On multiplying with 2 then

10 a - 8b = -4 -------(6)

On Subtracting (6) from (4) then

10a + 15b = 65

10a - 8b = -4

(-)

_____________

0 + 23 b = 69

______________

⇛ 23b = 69

⇛ b = 69/23

⇛ b =3

On Substituting the value of b in (5)

5a -4b= -2

⇛ 5a -4(3) = -2

⇛ 5a -12 = -2

⇛ 5a = -2+12

⇛ 5a = 10

⇛ a = 10/5

⇛ a = 2

Now we have

a = 2

⇛1/x = 2

⇛ x = 1/2

and

b = 3

⇛1/y = 3

⇛ y = 1/3

Answer :-The solution for the given problem is (1/2,1/3)

Check: If x = 1/2 and y = 1/3 then

LHS = (2/x)+(3/y)

= 2/(1/2)+3/(1/3)

= (2×2)+(3×3)

= 4+9

= 13

= RHS

LHS=RHS is true

and

LHS=(5/x)-(4/y)

⇛ 5/(1/2)- 4/(1/3)

⇛(5×2)-(4×3)

⇛ 10-12

⇛ -2

⇛RHS

LHS = RHS is true

Now,

X+Y

= (1/2) + (1/3)

Take the LCM of 2 and 3 is 6.

= (3+2)/6

= 5/6.

Answer 2

`x + y = (5)/(6)`

Explanation:

Let:

`u = (1)/(x) \\ v = (1)/(y)`

This allows us to manipulate the equations like we normally would.

`2u + 3v = 13 \\ 5u - 4v = - 2 \\ \\ + 8u + 12v = 52 \\ + 15u - 12v = - 6 \\ 23u = 46 \\ u = 2 \\ \\ 2(2) + 3v = 13 \\ 4 + 3v = 13 \\ 3v = 9 \\ v = 3`

Then, we return the values where they belong.

`u = (1)/(x) \\ 2 = (1)/(x) \\ x = (1)/(2) \\ \\ v = (1)/(y) \\ 3 = (1)/(y) \\ y = (1)/(3)`

Finally, we add:

`(1)/(2) + (1)/(3) = (3)/(6) + (2)/(6) = (5)/(6)`