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It appears you've uploaded an image containing a set of mathematical equations. These include ellipses and systems involving quadratic expressions. I'll walk through some of them and can provide solutions or explanations.

Let's start with the first few:

### 1. \(\frac{(x+6)^2}{16} + \frac{(y-5)^2}{16} = 1\)
This represents the equation of an ellipse centered at \((-6, 5)\) with equal semi-major and semi-minor axes of 4 units each (since \(\sqrt{16} = 4\)).

### 2. \(\frac{x^2}{300} + \frac{y^2}{300} = 1\)
This is a simplified ellipse centered at \((0,0)\) with axes of length \(\sqrt{300}\) in both directions, which is about 17.32 units.

### 3. \(\frac{(x-6)^2}{16} + \frac{(y-5)^2}{16} = 1\)
Another ellipse, but this one is centered at \((6, 5)\), also with axes of 4 units.

Would you like me to continue with the rest of the equations or focus on specific ones?

Here are some questions that follow up from this:

1. What properties define an ellipse in general?
2. How do we calculate the semi-major and semi-minor axes from the equation of an ellipse?
3. Can an ellipse have different axes, and how do you differentiate between them?
4. What other types of conic sections can be derived from these kinds of equations?
5. What happens if the right-hand side of the ellipse equation is not 1?

**Tip:** For ellipses, the larger denominator under \((x - h)^2\) or \((y - k)^2\) gives the length of the semi-major axis.

A set of equations, including ellipses and quadratic expressions such as (x+6)^2/16 + (y-5)^2/16 = 1, x^2/300 + y^2/300 = 1, and more complex forms like x^2 = 60(y+35).

Explore a detailed set of equations related to ellipses and quadratic expressions, including step-by-step analysis of equations like (x+6)^2/16 + (y-5)^2/16 = 1. Learn about the key mathematical concepts, formulas, and theorems that define conic sections and quadratic equations, suitable for high school students.

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