Question

Solve each of the following systems of equations by the method of cross-multiplication :

`5/(x + y) - 2/(x - y) = -1`

`15/(x + y) + 7/(x - y) = 10`

where `x != 0 and y != 0`

Answer 1

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

5u - 2v = -1

15u + 7v = 10

here

`a_1 = 5, b_1 = -2, c_1 = 1`

`a_2 = 15, b_2 = 7,c_2 = -10`

By cross multiplication, we get

`=> u/((-2)xx (-10)-1 xx7) = u/(5 xx (-10)-1 xx 15) = 1/(5xx7 - (-2) xx 15 )`

`=> u/(20 - 7) = (-v)/(-50 - 15) = 1/(35 + 30)`

`=> u/13 = (-v)/(-65) = 1/65`

`=> u/13 = v/65 = 1/65`

Now

`u/13 = 1/65`

`=> u = 13/65 = 1/5`

And

`v/65 = 1/65`

`=> v = 65/65 = 1`

Now

`u = 1/(x + y)`

`=> 1/(x + y) = 1/5`    ....(i)

And

`v = 1/(x - y)`

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

=> x - y = 1 .....(ii)

Adding equation (i) and (ii), we get

2x = 5 + 1

=> 2x = 6

`=> x = 6/2 = 3`

Adding x = 3 in eq 2

3 - y = 1

y = 3 -1

y = 2