We are given the system of equations:

\frac{2x}{3} - \frac{y}{2} = -\frac{1}{6} \tag{1} \frac{x}{2} + \frac{2y}{3} = 3 \tag{2}

To solve this system, let's start by eliminating the fractions to make the equations easier to work with.

Step 1: Eliminate fractions from both equations

For equation (1), we multiply both sides by 6 (the least common denominator of 3, 2, and 6) to clear the fractions: $6 \left( \frac{2x}{3} - \frac{y}{2} \right) = 6 \times -\frac{1}{6}$ Simplifying: 4x - 3y = -1 \tag{3}

For equation (2), we multiply both sides by 6 (the least common denominator of 2 and 3): $6 \left( \frac{x}{2} + \frac{2y}{3} \right) = 6 \times 3$ Simplifying: 3x + 4y = 18 \tag{4}

Now, we have the system of equations: 4x - 3y = -1 \tag{3} 3x + 4y = 18 \tag{4}

Step 2: Solve the system using substitution or elimination

Let's use the elimination method. We will eliminate one of the variables by multiplying both equations by suitable numbers.

Multiply equation (3) by 4 and equation (4) by 3:

4(4x - 3y) = 4(-1) \implies 16x - 12y = -4 \tag{5} 3(3x + 4y) = 3(18) \implies 9x + 12y = 54 \tag{6}

Now, add equations (5) and (6) together: $(16x - 12y) + (9x + 12y) = -4 + 54$ Simplifying: $25x = 50$ $x = \frac{50}{25} = 2$

Step 3: Substitute $x = 2$ into one of the original equations

Now that we know $x = 2$, substitute this value into equation (3) to solve for $y$: $4(2) - 3y = -1$ $8 - 3y = -1$ $-3y = -1 - 8 = -9$ $y = \frac{-9}{-3} = 3$

Final Solution:

The solution to the system of equations is: $x = 2, \quad y = 3$

Would you like any further clarification on the steps, or do you have any questions about this solution? Here's a list of related questions to explore:

1. How would you solve this system using the substitution method?
2. What happens if we use matrices to solve this system?
3. How can we check the solution by substituting $x = 2$ and $y = 3$ back into the original equations?
4. What are some different methods to solve linear equations with fractions?
5. How would the solution change if the constants in the equations were different?

Tip: When solving systems of linear equations, always look for the easiest way to eliminate fractions, such as multiplying through by the least common denominator (LCD). This simplifies calculations significantly.