You've provided equations from an image that need solving. Let's solve each equation one at a time.

Equation 1:

$y^2 - 2x^2 - 12x + 18y + 100 = 0$

Equation 2:

$3x^2 - y^2 - 12 = 0$

Equation 3:

$2x^2 - 3y^2 - 8x + 12y + 14 = 0$

Let's start by solving each of these equations. We'll handle them one by one to find any specific points or solutions. We'll begin with Equation 1 and proceed to the others.Here are the solutions for each equation in terms of $y$ as a function of $x$:

Equation 1:

$y = -\sqrt{2x^2 + 12x - 19} - 9 \quad \text{or} \quad y = \sqrt{2x^2 + 12x - 19} - 9$

Equation 2:

$y = -\sqrt{3x^2 - 12} \quad \text{or} \quad y = \sqrt{3x^2 - 12}$

Equation 3:

$y = 2 - \frac{\sqrt{6x^2 - 24x + 78}}{3} \quad \text{or} \quad y = \frac{\sqrt{6x^2 - 24x + 78}}{3} + 2$

These are the expressions for $y$ as functions of $x$ for each equation. If you need to find specific points or intersections, please specify further or ask for a graphical analysis. Would you like to proceed with any specific calculations or have further questions on these equations?

Here are some related questions you might find interesting:

1. How can you find the intersections of these equations?
2. What are the graphical representations of these equations?
3. How can you determine the range of $x$ values for which these solutions are real?
4. Could these equations represent conic sections, and if so, what types?
5. What is the significance of the discriminant in each solution?

Tip: When dealing with square roots in equations, ensure the expressions under the square roots are non-negative to maintain real solutions.