Equations Reducible to a Pair of Linear Equations in Two Variables

In the earlier concepts, we studied three methods to solve the Linear Equations, where we were directly provided with a linear equation which was in a standard form, i.e. `a_1x+b_1y+c_1=0` and `a_2x+b_2+c_2=0`. Here, in this concept, we are required to reduce the given question into a proper linear equation and then solve it.

Example- `1/"x-1" + 2/"y-2" = 2` and `3/"x-1" - 3/"y-2" = 1`

First, we need to reduce this equation in `a_1x+b_1y+c=0` and `a_2x+b_2y+c_2=0` 

For that, let's take `1/"x-1"` = m and `1/"y-2"` =n 

So the equation becomes like, `m+2n=2` ....eq1
and `3m-3n=1` ....eq2
By solving further, we get `m=2-2n`, substituting this in eq2
`3(2-2n)-3n=1`
`6-6n-3n=1`
`-9n= 1-6`

`n= (-5)/-9`  i.e. `5/9`

Substitute `n=5/9` into eq1

`m+2(5/9)=2`

`m+ 10/9= 2`

`m= 2-10/9`

`m= 8/9`

Now we will resubstitute the values of m and n in the original equations.

`1/"x-1"= 8/9`

`8x-8= 9`
`8x=17`

`x=17/8`

And `1/"y-2"= 5/9`

`5y-10= 9`
`5y= 9+10`

`y= 19/5`