Find the ratio of x to y, given the 2 equations.

Question

6/y + 5/x = 31

10/y -4/x = 31

Answer 1

(6/y) + (5/x) = 31 can be simplified and written as 6x + 5y = 31xy

(10/y) - (4/x) = 31 can be simplified and written as 10x-4 y = 31xy

from the above two equtions:

6x + 5y = 10x - 4y

or (x/y) = 9/4

Answer 2

6/y + 5/x = 31 and 10/y - 4/x = 31

⇒ 6/y + 5/x = 10/y - 4/x

⇒ 6/y + 5/x + 4/x = 10/y - 4/x + 4/x

⇒ 6/y + 9/x = 10/y

⇒ 6/y + 9/x - 6/y = 10/y - 6/y

⇒ 9/x = 4/y

⇒ 9/x * x/4 = 4/y * x/4

⇒ 9/4 = x/y

Answer 3

It is not necessary to solve separately for x and y to find their ratio. You can use the symmetry of the equations to shortcut the work.

Multiply the two equations by x:

6/y + 5/x = 31 -> 6x/y + 5 = 31x

10/y -4/x = 31 -> 10x/y - 4 = 31x

Subtract the second from the first equation, that will cancel the terms 31x:

-4x/y + 9 = 0

Solve for x/y: x/y = 9/4