For solving each pair of equations, in this exercise, use the method of elimination by equation coefficiients :
2x - 3y -3 = 0
`(2x)/(3) + 4y + (1)/(2) = 0`

Answer

To solve the given pair of equations using the method of elimination by equation coefficients, we will follow these steps:

### Given Equations:
1. \( 2x - 3y - 3 = 0 \)
2. \( \frac{2x}{3} + 4y + \frac{1}{2} = 0 \)

### Step 1: Rearranging the Equations
First, let's rearrange both equations in standard form \( Ax + By + C = 0 \).

1. For the first equation:
\[ 2x - 3y - 3 = 0 \implies 2x - 3y = 3 \]

2. For the second equation, we will eliminate the fractions by multiplying through by 6 (the least common multiple of the denominators 3 and 2):
\[ 6 \left( \frac{2x}{3} + 4y + \frac{1}{2} \right) = 0 \implies 4x + 24y + 3 = 0 \implies 4x + 24y = -3 \]

### Step 2: Writing the Equations in Standard Form
Now we have the equations:
1. \( 2x - 3y = 3 \) (Equation 1)
2. \( 4x + 24y = -3 \) (Equation 2)

### Step 3: Making Coefficients of \( x \) Equal
To eliminate \( x \), we will multiply the first equation by 2 so that the coefficients of \( x \) in both equations become equal:
\[ 2(2x - 3y) = 2(3) \implies 4x - 6y = 6 \quad \text{(Equation 3)} \]

### Step 4: Subtracting the Equations
Now we will subtract Equation 2 from Equation 3:
\[ (4x - 6y) - (4x + 24y) = 6 - (-3) \]
This simplifies to:
\[ -6y - 24y = 6 + 3 \implies -30y = 9 \]

### Step 5: Solving for \( y \)
Now we can solve for \( y \):
\[ y = \frac{9}{-30} = -\frac{3}{10} \]

### Step 6: Substituting \( y \) Back to Find \( x \)
Now we substitute \( y = -\frac{3}{10} \) back into one of the original equations to find \( x \). We can use Equation 1:
\[ 2x - 3\left(-\frac{3}{10}\right) = 3 \]
This simplifies to:
\[ 2x + \frac{9}{10} = 3 \]
Subtracting \( \frac{9}{10} \) from both sides:
\[ 2x = 3 - \frac{9}{10} = \frac{30}{10} - \frac{9}{10} = \frac{21}{10} \]
Now, divide by 2:
\[ x = \frac{21}{20} \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = \frac{21}{20}, \quad y = -\frac{3}{10} \]