The exponential equation `12^(3x - y) = 72^(x+1)` has to be solved for x and y.

One of the solutions of this equation can be determined as follows:

`12^(3x - y) = 72^(x+1)`

`12 = 3*4 = 3*2^2`

`72 = 9*8 = 3^2*2^3`

The equation can be rewritten as:

`(3*2^2)^(3x - y) = (3^2*2^3)^(x+1)`

`3^(3x - y)*2^(2*(3x-y)) = 3^(2*(x+1))*2^(3*(x+1))`

=> `3^(3x - y)*2^(6x - 2y) = 3^(2x+2)*2^(3x+3)`

Equating the exponents of 3, 3x - y = 2x + 2

x = 2 + y

Equating the exponents of 2, 6x - 2y = 3x + 3

3x = 3 + 2y

Substitute x = 2 + y

6+3y = 3 + 2y

y = -3

x = -1

**The solution of the equation is x = -1 and y = -3.**

You should also note that the general solution is,

`y= 3x - (ln72/ln12)x - (ln72/ln12) `

Also, since the equation is one with two variables, there are an infinite number of solutions.

Given,

` 12^(3x - y) = 72^(x+1)`

` sol:-`

` 12^(3x - y) = 72^(x+1) `

As the above equation has **two unknowns x, y**. In order to get values of **two unknowns** , we must have **atleast two equations** to solve but we have **only one**

applying Log on both sides, we get

` log(12^(3x-y))=log(72^(x+1))`

This is of the form

`log (a^x)= x*log (a)`

` => log(12^(3x-y))=log(72^(x+1)) `

` => (3x-y) * log (12)= (x+1)* log(72)`

is the **linear equation** (which is represented with a straight line when ploted on a graph).

Now from the above equation using trail and error method , we get **infinite number** of possibilities of values of (x,y) belongs to **Real numbers**

for example

we can find the values of (x,y ) which cuts the co-ordinate axis

let x=0 then value of y is

` y= (-log(72)/log(12))`

if y=0 then value of x is

` x=log(72)/(3*log(12)-log(72))`

The linear graph is plotted and can be seen in the attached files :)