Let's solve the given equation:

\[
\frac{x^2}{\frac{35}{3}} + \frac{y^2}{\frac{35}{8}} = 1
\]

First, simplify the equation by rewriting it in a more standard form:

\[
\frac{3x^2}{35} + \frac{8y^2}{35} = 1
\]

Now, multiply the entire equation by 35 to eliminate the fractions:

\[
3x^2 + 8y^2 = 35
\]

This is the simplified equation of the ellipse. The task of solving this equation involves identifying its key properties, such as its center, axes lengths, and orientation, rather than solving for specific \(x\) and \(y\) values unless additional conditions are provided. Here's the breakdown:

### Step 1: Identify the Center
Since there are no \(x\) or \(y\) terms being added or subtracted, the center of the ellipse is at the origin \((0, 0)\).

### Step 2: Determine the Axes
The general form of the ellipse equation is:

\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]

Comparing with our equation \(3x^2 + 8y^2 = 35\):

- \(a^2 = \frac{35}{3}\)
- \(b^2 = \frac{35}{8}\)

### Step 3: Calculate the Lengths of the Axes
- \(a = \sqrt{\frac{35}{3}} = \frac{\sqrt{105}}{3}\)
- \(b = \sqrt{\frac{35}{8}} = \frac{\sqrt{280}}{8} = \frac{\sqrt{35}}{2\sqrt{2}}\)

So, the semi-major axis is along the \(y\)-axis since \(b > a\), and the lengths are:

- Semi-major axis length = \(2b = 2 \times \frac{\sqrt{35}}{\sqrt{8}} = \sqrt{\frac{35}{2}}\)
- Semi-minor axis length = \(2a = 2 \times \frac{\sqrt{35}}{3\sqrt{3}} = \frac{2\sqrt{35}}{3}\)

### Conclusion
The ellipse represented by the equation \(3x^2 + 8y^2 = 35\) has its center at \((0, 0)\) and is elongated along the y-axis with axes lengths as calculated above.

Would you like more details or have any questions? Here are five related questions you might find helpful:

1. How would the equation change if the ellipse were rotated by a certain angle?
2. What is the relationship between the coefficients in the equation and the orientation of the ellipse?
3. How can you determine the foci of an ellipse given its equation?
4. How do you graph an ellipse from its equation?
5. What are some real-world applications of ellipses?

**Tip:** When dealing with ellipses, knowing how to rewrite the equation in a standard form helps in identifying key properties like the center, axes, and orientation.

Learn how to solve the ellipse equation 3x^2 + 8y^2 = 35 step-by-step. Understand the properties of ellipses, including the center, axes lengths, and orientation. Suitable for high school students studying conic sections.

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