Determine whether each equation defines y as a function of x If the equation does define y as a function of x, enter y in terms of x in the box provided. If the equation does not define y as a function of x, enter none in the box provided. 7x ^ 2 + 8y ^ 2 = 6 y none 29x ^ 2 + 5y = 2; y = Box Sigma Σ

Let's carefully determine whether each equation defines \(y\) as a function of \(x\) and find \(y\) in terms of \(x\) when possible.

### Equation 1: \(7x^2 + 8y^2 = 6y\)

To determine if \(y\) is a function of \(x\), we need to check if for every \(x\), there is exactly one corresponding value of \(y\).

1. Rewrite the equation to focus on \(y\):
   \[
   8y^2 - 6y + 7x^2 = 0
   \]

2. This is a quadratic equation in terms of \(y\). A quadratic equation can have 0, 1, or 2 solutions for \(y\) depending on the value of the discriminant. The quadratic equation can be written in the standard form:
   \[
   ay^2 + by + c = 0
   \]
   where \(a = 8\), \(b = -6\), and \(c = 7x^2\).

3. Compute the discriminant \(\Delta\):
   \[
   \Delta = b^2 - 4ac = (-6)^2 - 4 \cdot 8 \cdot 7x^2 = 36 - 224x^2
   \]

4. Since the discriminant \(36 - 224x^2\) can be positive, zero, or negative depending on the value of \(x\), the equation can have two different solutions, one solution, or no real solution for \(y\).

Because there can be more than one \(y\) value for a given \(x\), **Equation 1 does not define \(y\) as a function of \(x\)**.

**Answer for Equation 1: `none`**

### Equation 2: \(29x^2 + 5y = 2\)

Let's solve for \(y\) to check if it is a function of \(x\).

1. Rewrite the equation:
   \[
   5y = 2 - 29x^2
   \]

2. Solve for \(y\):
   \[
   y = \frac{2 - 29x^2}{5}
   \]

This is a linear equation, and for each \(x\), there is exactly one value of \(y\). Thus, **Equation 2 defines \(y\) as a function of \(x\)**.

**Answer for Equation 2: \(y = \frac{2 - 29x^2}{5}\)**

### Conclusion:
1. **Equation 1: \(7x^2 + 8y^2 = 6y\)**  
   Answer: **none** (does not define \(y\) as a function of \(x\)).

2. **Equation 2: \(29x^2 + 5y = 2\)**  
   Answer: \(y = \frac{2 - 29x^2}{5}\) (defines \(y\) as a function of \(x\)).

Would you like more details or have any questions?

#### Related Questions:
1. How does the discriminant determine the number of solutions for a quadratic equation?
2. What are some examples of equations that do not define functions?
3. How can you graphically check if an equation defines a function?
4. What is the significance of rewriting equations to solve for \(y\)?
5. How can linear equations always define functions?

**Tip:** Quadratic equations in \(y\) can have up to two different solutions for each \(x\), so they often do not define \(y\) as a function of \(x\).

Understand how to determine if an equation defines y as a function of x with detailed examples and solutions. This problem covers functions, quadratic equations, and linear equations, suitable for high school students.

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