Could somebody help me solve the following equation

Question

Could somebody help me solve the following equation for x and y:

$7^{2n - 3}\cdot 7^{3}\cdot 7^{n + 3} = 7^{xn + y + 1}$

I reached a step where:

3(n + 1) = xn + y - 1

but can't figure out how to proceed :-(

Answer 1

You're almost there.

You seem to be aware that you can multiply powers by adding the exponents:

Hence $$ 7^{2n-3} \times 7^3 \times 7^{n+3} = 7^{(2n-3) + 3 + (n+3)} = 7^{3n+3} $$

(I think you are that far along, since you got to your next formula).

Don't bother factoring $3n+3$ - it doesn't help.

Now, how would you choose $x$ and $y$ so that $xn + y - 1 = 3n+3$?

Notice that $n$ appears in only one term on left and right, so let $x = 3$ and you're left with $3n + y - 1 = 3n + 3$. Can you take it the rest of the way?

(ETA: You answered your own question while I was typing this!)

Answer 2

I think I got the response: 3n + 3 = xn + y - 1

3n = xn if x = 3

3 = y - 1 if y = 4