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.
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.
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.
` 12^(3x - y) = 72^(x+1)`
` 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
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
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
The linear graph is plotted and can be seen in the attached files :)