What is the value of x and y given that 12^(3x - y) = 72^(x+1)

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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.

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

I am sure that this is the answer that is expected, but what a terrible question. There are an infinite number of solutions. The general solution is `y=3x-(ln72)/(ln12)x-ln(72)/(ln12) `

or approximately y=1.278942946x-1.721057054.

When x=-1, y is indeed -2 as one of the infinite solutions.

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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

Please note that the values given for x and y, as noted in the response, is only one of the infinite number of solutions.

kseddy123's profile picture

kseddy123 | College Teacher | (Level 1) Assistant Educator

Posted on

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 :)

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