Let's solve the system of linear equations:

\[
\begin{aligned}
4x - \frac{1}{3}y &= \frac{2}{3} \quad \text{(Equation 1)} \\
\frac{1}{2}x + \frac{1}{5}y &= -11 \quad \text{(Equation 2)}
\end{aligned}
\]

### Step 1: Eliminate fractions by multiplying both equations by suitable numbers

For **Equation 1**, multiply by 3 to eliminate the fraction:

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

This simplifies to:

\[
12x - y = 2 \quad \text{(Equation 3)}
\]

For **Equation 2**, multiply by 10 to eliminate the fractions (since 10 is the least common multiple of 2 and 5):

\[
10\left(\frac{1}{2}x + \frac{1}{5}y\right) = 10(-11)
\]

This simplifies to:

\[
5x + 2y = -110 \quad \text{(Equation 4)}
\]

### Step 2: Use the elimination method

Let's eliminate \( y \). Multiply **Equation 3** by 2 so that the coefficients of \( y \) are the same in both equations:

\[
2(12x - y) = 2(2)
\]

This simplifies to:

\[
24x - 2y = 4 \quad \text{(Equation 5)}
\]

Now subtract **Equation 4** from **Equation 5**:

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

Simplify:

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

\[
19x = 114
\]

### Step 3: Solve for \( x \)

Divide by 19:

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

### Step 4: Substitute \( x = 6 \) into one of the original equations

Substitute \( x = 6 \) into **Equation 3**:

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

Simplify and solve for \( y \):

\[
72 - y = 2
\]

Subtract 72 from both sides:

\[
-y = 2 - 72 = -70
\]

Multiply by \(-1\):

\[
y = 70
\]

### Solution

The solution to the system of equations is:

\[
x = 6, \quad y = 70
\]

Would you like more details or have any questions?

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Here are 5 related questions you might explore:
1. How do you handle systems of equations with fractions efficiently?
2. What other methods can be used to solve this system of equations?
3. How can you interpret the solution graphically?
4. What happens if you substitute incorrect values into an equation?
5. How can you verify the solution using substitution?

**Tip:** When dealing with fractions in equations, multiplying by the least common multiple (LCM) can simplify the problem and help avoid errors.

Learn how to solve the system of linear equations 4x - 1/3y = 2/3 and 1/2x + 1/5y = -11 step-by-step with detailed solutions. This problem covers linear equations with fractions, utilizing the elimination method to find the values of x and y. Suitable for high school students.

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