Question

Solve the following system of equations by the method of cross-multiplication:

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

`38/(x + y) + 21/(x - y) = 9`

Answer 1

Let `1/(x +y) = u and 1/(x -y ) = v` Then the given system of equations become

`57u + 6v = 5 => 57u + 6v - 5 = 0`

`38u + 21v = 9 => 38u + 21v - 9 = 0`

Here

`a_1 = 57, b_1 = 6, c_1 = -5`

`a_2 = 38, b_2 = 21, c_2 = -9`

By cross multiplication, we have

`=> u/(-54 + 105) = (-v)/(-513 + 190) = 1/(1193 - 228)`

`=> u/51 = (-v)/(-323) = 1/969`

`=> u/51 = v/323 = 1/969`

Now

`u/51= 1/969`

`=> u = 51/969`

`=> u = 1/19`

And

`v/(323) = 1/969`

`=> v = 323/969`

`=> v = 1/3`

Now

`u = 1/(x + y)`

`=> 1/(x + y)  =1/ 19`

`=>x + y = 19` ....(i)

And

`v = 1/(x - y)` 

`=> 1/(x -y) = 1/3`

`=> x - y = 3` ...(ii)

Now adding equation i and ii

we get x = 11

And after substituting the value of x in equation (ii)

we get y = 8

Hence  the value of x = 11 and y = 8.