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You provided a system of equations to solve. The system is:

\[
4x - \frac{1}{3}y = \frac{2}{3}
\]
\[
\frac{1}{2}x - \frac{1}{5}y = -11
\]

To solve this, we can use either substitution or elimination. I will use the elimination method after eliminating the fractions.

### Step 1: Eliminate Fractions
1. Multiply the first equation by 3 to eliminate the denominator:

\[
3(4x - \frac{1}{3}y) = 3 \times \frac{2}{3}
\]
\[
12x - y = 2
\]

2. Multiply the second equation by 10 to eliminate the fractions:

\[
10(\frac{1}{2}x - \frac{1}{5}y) = 10 \times -11
\]
\[
5x - 2y = -110
\]

Now the system is:

\[
12x - y = 2
\]
\[
5x - 2y = -110
\]

### Step 2: Solve the System
We can use elimination to solve for \(y\).

1. Multiply the first equation by 2 to align the \(y\)-terms:

\[
2(12x - y) = 2 \times 2
\]
\[
24x - 2y = 4
\]

Now the system is:

\[
24x - 2y = 4
\]
\[
5x - 2y = -110
\]

2. Subtract the second equation from the first:

\[
(24x - 2y) - (5x - 2y) = 4 - (-110)
\]
\[
24x - 5x = 4 + 110
\]
\[
19x = 114
\]

Solving for \(x\):

\[
x = \frac{114}{19} = 6
\]

### Step 3: Solve for \(y\)
Substitute \(x = 6\) into one of the original equations. Using \(12x - y = 2\):

\[
12(6) - y = 2
\]
\[
72 - y = 2
\]
\[
y = 72 - 2 = 70
\]

### Final Solution:
The solution to the system is:
\[
x = 6, \quad y = 70
\]

Would you like further clarification on any step or additional details?

### Related Questions:
1. How would you solve the system using the substitution method?
2. Can the system be solved graphically, and how would the solution appear on a graph?
3. What happens if you change the second equation's constant term?
4. How would the solution change if the first equation had different coefficients?
5. What are some applications of solving linear systems like these?

### Tip:
When solving systems with fractions, eliminating them early on can make the process easier and reduce the chance of errors.

Solve the following system of equations:

4x - (1/3)y = 2/3

(1/2)x - (1/5)y = -11

This page provides a detailed step-by-step solution to solve the system of equations 4x - (1/3)y = 2/3 and (1/2)x - (1/5)y = -11 using the elimination method. Learn how to handle fractions and solve for both variables, suitable for students in Grades 9-11.

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