Question

Solve the equation $9^{2y-1}={27}^y$. Check your solution.

Short answer

The solution for the equation is$y=2$.

Step-by-step solution

Step 1. Write down the given expression.

The given equation is $9^{2y-1}={27}^y$.

Step 2. Simplify the equation.

The given equation $9^{2y-1}={27}^y$is solved as:

$\begin{array}{l}{9}^{2y-1}={27}^y\mathrm{....}\left(\text{Given}\right)\\ {\left(3^2\right)}^{2y-1}={\left(3^3\right)}^y\\ {\left(3\right)}^{4y-2}={\left(3\right)}^{3y}\\ 4y-2=3y\\ y=2\end{array}$

Step 3. Verify the result for obtained value.

Plugging $y=2$in given equation and verify the result.

$\begin{array}{l}{9}^{2y-1}={27}^y\mathrm{....}\left(\text{Given}\right)\\ 9^{2\left(2\right)-1}={27}^{\left(2\right)}\mathrm{....}\left(\text{Plugging}y=2\right)\\ 9^3={27}^{\left(2\right)}\\ 729=\mathrm{729....}\left(\text{True}\right)\end{array}$

Hence the solution is verified for $y=2$.

Step 4. Conclusion.

The solution for the given equation is $y=2$.