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. And  29x ^ 2 + 5y = 2; y

To determine if an equation defines \(y\) as a function of \(x\), we need to check if, for each value of \(x\), there is exactly one corresponding value of \(y\). If there is more than one value of \(y\) for a given \(x\), then \(y\) is not a function of \(x\).

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

Let's analyze this equation:

1. Rewrite the equation to make it easier to analyze:
   \[
   8y^2 - 6y + 7x^2 = 0
   \]

2. This is a quadratic equation in terms of \(y\):
   \[
   ay^2 + by + c = 0 \quad \text{where } a = 8, \, b = -6, \, c = 7x^2
   \]

3. A quadratic equation can have either two solutions, one solution, or no solution for \(y\) depending on the discriminant:
   \[
   \Delta = b^2 - 4ac
   \]
   Here,
   \[
   \Delta = (-6)^2 - 4(8)(7x^2) = 36 - 224x^2
   \]

4. Since the discriminant (\(36 - 224x^2\)) can be positive, zero, or negative depending on \(x\), this quadratic equation may have two solutions, one solution, or no solution for \(y\). This means that for some values of \(x\), there could be two different \(y\) values, which means \(y\) is **not a function** of \(x\).

Thus, **Equation 1 does not define \(y\) as a function of \(x\)**. The answer is **none**.

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

Let's solve for \(y\) in terms of \(x\):

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

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

3. This is a linear equation in terms of \(y\), and for every \(x\), there is exactly one corresponding \(y\).

Thus, **Equation 2 does define \(y\) as a function of \(x\)**. The answer is:

\[
y = \frac{2 - 29x^2}{5}
\]

### Summary:
1. For \(7x^2 + 8y^2 = 6y\), the answer is **none** (does not define \(y\) as a function of \(x\)).
2. For \(29x^2 + 5y = 2\), the answer is \(y = \frac{2 - 29x^2}{5}\) (defines \(y\) as a function of \(x\)).

Would you like more details or have any questions?

#### Related Questions:
1. How do you determine if a quadratic equation defines \(y\) as a function of \(x\)?
2. What are the characteristics of an equation that defines a function?
3. How does the discriminant affect the number of solutions to a quadratic equation?
4. What are some examples of equations that do not define \(y\) as a function of \(x\)?
5. How do you graph equations to visually determine if \(y\) is a function of \(x\)?

**Tip:** When checking if an equation defines a function, always solve for \(y\) in terms of \(x\) and see if each \(x\) corresponds to exactly one \(y\).

Explore whether y is defined as a function of x through quadratic equations. Analyze equations like 7x^2 + 8y^2 = 6y and 29x^2 + 5y = 2 to understand the conditions under which y can be a function of x. Suitable for Grades 7-9.

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