To solve for $x$ in the given triangle, we need to use trigonometric relationships. Here's the step-by-step process:

1. Identify the given angles and sides:

   • Angle opposite the unknown side ($x$): $27^\circ$
   • Angle adjacent to the unknown side: $64^\circ$
   • Side opposite the angle $64^\circ$: $97 + x$

2. Use the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ For angle $27^\circ$: $\tan(27^\circ) = \frac{97 + x}{x}$

3. Solve for $x$: $x \tan(27^\circ) = 97 + x$

4. Isolate $x$: $x \tan(27^\circ) - x = 97$ $x (\tan(27^\circ) - 1) = 97$ $x = \frac{97}{\tan(27^\circ) - 1}$

5. Calculate the value of $x$: Using the tangent of $27^\circ$: $\tan(27^\circ) \approx 0.5095$ Therefore: $x = \frac{97}{0.5095 - 1}$ $x = \frac{97}{-0.4905}$ $x \approx -197.77$

It seems we have a negative value, which suggests that there might be an error in interpretation or calculation. Re-evaluating the setup might be necessary.

Alternatively, another approach using sine or cosine law could be attempted, but as it stands, the direct tangent approach suggests rechecking the problem context for any missing pieces.

Would you like more details, or do you have any questions? Here are 5 questions you might ask:

1. Can you explain how to solve for $x$ using a different trigonometric function?
2. What are some common mistakes to avoid in trigonometry problems?
3. How can I verify the accuracy of my trigonometric calculations?
4. What other methods can be used to solve triangles besides trigonometry?
5. Can you provide an example of solving a right triangle with given sides?

Tip: Always double-check the problem's context and given values to ensure accuracy in your calculations.