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 $x$.