I have displayed the image you uploaded. Let's solve the given rational equations using the least common denominator (LCD) method. I'll begin by extracting each equation and solving them step by step. Here are the equations:

1. $\frac{5}{x} + \frac{4}{5x} = \frac{3}{x}$
2. $\frac{3}{2y} = \frac{5}{y^2}$
3. $\frac{x}{y-2} = \frac{2}{y+3}$

Let's start by solving the first equation:

1. $\frac{5}{x} + \frac{4}{5x} = \frac{3}{x}$

• The least common denominator (LCD) is $5x$.
• Multiply both sides by the LCD to eliminate the denominators.

I'll perform the calculations and provide the solution next.It seems there are no real solutions for the first equation $\frac{5}{x} + \frac{4}{5x} = \frac{3}{x}$. This likely means that no value of $x$ satisfies the equation without introducing contradictions, such as division by zero.

Now, let's proceed to solve the second equation:

2. $\frac{3}{2y} = \frac{5}{y^2}$

• The least common denominator (LCD) is $2y^2$.
• I'll multiply both sides by the LCD and solve the equation.The solution to the second equation $\frac{3}{2y} = \frac{5}{y^2}$ is:

$y = \frac{10}{3}$

Let's now move on to the third equation:

3. $\frac{x}{y-2} = \frac{2}{y+3}$

I'll solve this equation by cross-multiplying both sides.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can still assist with explanations or walk you through how to solve the equations manually if you'd like. Let me know how you'd like to proceed!