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The question asks which system of trigonometric equations has a solution with an x-coordinate of \( \frac{\pi}{4} \). Let's analyze each option by evaluating the functions at \( x = \frac{\pi}{4} \) and checking if both equations in the system yield the same value of \( y \).

1. **Option a:**
   - \( y = \sin(x) \), so \( y = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).
   - \( y = \cos(7x) \), so \( y = \cos\left(7 \cdot \frac{\pi}{4}\right) = \cos\left(\frac{7\pi}{4}\right) = \cos\left(2\pi - \frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).
   - Both values are equal, so **option a** is a solution.

2. **Option b:**
   - \( y = \sin(x) \), so \( y = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).
   - \( y = \cos(3x) \), so \( y = \cos\left(3 \cdot \frac{\pi}{4}\right) = \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \).
   - The values are not equal, so **option b** is not a solution.

3. **Option c:**
   - \( y = \sin(x) \), so \( y = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).
   - \( y = \cos(5x) \), so \( y = \cos\left(5 \cdot \frac{\pi}{4}\right) = \cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \).
   - The values are not equal, so **option c** is not a solution.

4. **Option d:**
   - \( y = \sin(x) \), so \( y = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).
   - \( y = \cos(9x) \), so \( y = \cos\left(9 \cdot \frac{\pi}{4}\right) = \cos\left(\frac{9\pi}{4}\right) = \cos\left(2\pi + \frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).
   - Both values are equal, so **option d** is also a solution.

### Final Answer:
The correct systems of equations are **a** and **d**.

Would you like any further clarification or details?

Here are some follow-up questions:
1. What are the periodic properties of sine and cosine functions?
2. How can we use reference angles to simplify trigonometric evaluations?
3. What is the significance of the angle \( \frac{\pi}{4} \) in trigonometry?
4. How do sine and cosine functions relate for supplementary angles?
5. Can you solve a trigonometric system graphically?

**Tip**: For trigonometric equations, periodicity can often give you more than one solution for a given value of \( x \).

Which of the following systems of trigonometric equations have a solution with an x-coordinate of π/4?

Discover which systems of trigonometric equations have a solution with x-coordinate π/4. Learn key concepts in trigonometry, including sine and cosine functions and periodicity. Suitable for students in Grades 10-12.

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