We are given the following system of equations:

$\frac{1}{2}x - \frac{1}{3}y = -5$ $2x - \frac{1}{5}y = \frac{2}{5}$

Let's solve this step-by-step using substitution or elimination. First, we can eliminate the fractions by multiplying each equation by the least common denominator (LCD) of the fractions in each.

Step 1: Eliminate fractions in the first equation

The first equation is: $\frac{1}{2}x - \frac{1}{3}y = -5$ The LCD of 2 and 3 is 6. Multiply the entire equation by 6: $6 \times \left(\frac{1}{2}x - \frac{1}{3}y\right) = 6 \times (-5)$ This simplifies to: $3x - 2y = -30 \quad \text{(Equation 1)}$

Step 2: Eliminate fractions in the second equation

The second equation is: $2x - \frac{1}{5}y = \frac{2}{5}$ Multiply the entire equation by 5 to eliminate the fractions: $5 \times \left(2x - \frac{1}{5}y\right) = 5 \times \frac{2}{5}$ This simplifies to: $10x - y = 2 \quad \text{(Equation 2)}$

Step 3: Solve the system of equations

We now have the system: $3x - 2y = -30 \quad \text{(Equation 1)}$ $10x - y = 2 \quad \text{(Equation 2)}$

We can use substitution or elimination to solve this system. Let's use elimination by first multiplying Equation 2 by 2 so that the coefficients of $y$ match:

$2(10x - y) = 2(2)$ $20x - 2y = 4 \quad \text{(Equation 3)}$

Now subtract Equation 1 from Equation 3:

$(20x - 2y) - (3x - 2y) = 4 - (-30)$ $20x - 3x = 4 + 30$ $17x = 34$ $x = 2$

Step 4: Substitute $x = 2$ into one of the original equations

Substitute $x = 2$ into Equation 2: $10(2) - y = 2$ $20 - y = 2$ $y = 18$

Final Solution:

The solution to the system is: $x = 2, \quad y = 18$

Would you like more details on any part of the process or have any questions?

Here are some related questions for further exploration:

1. How do you solve a system of equations using the substitution method?
2. What is the process of solving a system of equations with fractions?
3. How does elimination work for solving systems of equations?
4. What happens when you have no solution or infinitely many solutions to a system?
5. Can systems of equations be solved graphically? How?

Tip: When dealing with fractions in equations, multiplying through by the least common denominator is a quick way to simplify the system.