Solve the system of equations: 3x + 2y = 11, 2x − y = 1. Then, solve 5^x = 243.

Given the equations:

1. $3x + 2y = 11$
2. $2x - y = 1$

From equation (2), we can isolate $y$:

$$y = 2x - 1$$

$$3x + 2(2x - 1) = 11$$ $$3x + 4x - 2 = 11$$ $$7x - 2 = 11$$ $$7x = 11 + 2$$ $$7x = 13$$ $$x = \frac{13}{7}$$

$$y = 2\left(\frac{13}{7}\right) - 1$$ $$y = \frac{26}{7} - \frac{7}{7}$$ $$y = \frac{19}{7}$$

$$x = \frac{13}{7} \approx 1.8571... \approx 1.86$$ $$y = \frac{19}{7} \approx 2.7142... \approx 2.71$$

The solutions are $\boxed{\mathbf{x = 1.86, y = 2.71}}$.

To solve for $x$, take the logarithm of both sides. Using the natural logarithm (ln):

$$\ln(5^x) = \ln(243)$$

Apply the logarithm property $\ln(a^b) = b \ln(a)$:

$$x \ln(5) = \ln(243)$$

$$x = \frac{\ln(243)}{\ln(5)}$$

Using a calculator:

$$\ln(243) \approx 5.49306$$ $$\ln(5) \approx 1.60944$$ $$x \approx \frac{5.49306}{1.60944} \approx 3.41299...$$

Rounding to two decimal places:

$$x \approx 3.41$$

The value of $x$ is $\boxed{\mathbf{3.41}}$.