Solve:
`{:((x)/(3) + (y)/(2) = 3),(x - 2y = 2):}`

Answer

To solve the system of equations

\[ \frac{x}{3} + \frac{y}{2} = 3 \]
\[ x - 2y = 2 \]

we will follow these steps:

### Step 1: Rewrite the first equation in standard form
To eliminate the fractions in the first equation, we find the least common multiple (LCM) of the denominators (3 and 2), which is 6. We multiply the entire equation by 6:

\[ 6 \left(\frac{x}{3}\right) + 6 \left(\frac{y}{2}\right) = 6 \cdot 3 \]

This simplifies to:

\[ 2x + 3y = 18 \]

Now we have the first equation in standard form:

\[ 2x + 3y = 18 \quad \text{(Equation 1)} \]

### Step 2: Write the second equation
The second equation is already in standard form:

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

### Step 3: Solve for one variable
We will solve Equation 2 for \(x\):

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

### Step 4: Substitute into the first equation
Now we substitute the expression for \(x\) from Equation 3 into Equation 1:

\[ 2(2 + 2y) + 3y = 18 \]

### Step 5: Simplify and solve for \(y\)
Expanding the left side:

\[ 4 + 4y + 3y = 18 \]

Combine like terms:

\[ 4 + 7y = 18 \]

Now, isolate \(y\) by subtracting 4 from both sides:

\[ 7y = 14 \]

Now, divide by 7:

\[ y = 2 \]

### Step 6: Substitute back to find \(x\)
Now that we have \(y\), we substitute it back into Equation 3 to find \(x\):

\[ x = 2 + 2(2) \]

This simplifies to:

\[ x = 2 + 4 = 6 \]

### Final Solution
Thus, the solution to the system of equations is:

\[ x = 6, \quad y = 2 \]

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