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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.

Solve the system of equations: (1/2)x - (1/3)y = -5 and 2x - (1/5)y = 2/5.

Learn how to solve the system of equations (1/2)x - (1/3)y = -5 and 2x - (1/5)y = 2/5 using elimination and substitution. This problem involves handling fractions and linear algebra concepts, suitable for students in Grades 9-12.

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