Solve the following pairs of linear (simultaneous) equation using method of elimination by subsitution:
`(3x)/(2) - (5y)/(3) + 2 = 0`
`(x)/(3) + (y)/(2) = 2(1)/(6)`

Answer

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

### Step 1: Convert the equations to standard form
The given equations are:
1. \(\frac{3x}{2} - \frac{5y}{3} + 2 = 0\)
2. \(\frac{x}{3} + \frac{y}{2} = \frac{2}{6}\)

First, we will eliminate the fractions by finding a common denominator.

**For the first equation:**
- The least common multiple (LCM) of 2 and 3 is 6.
- Multiply the entire equation by 6 to eliminate the fractions:

\[ 6 \left(\frac{3x}{2}\right) - 6 \left(\frac{5y}{3}\right) + 6(2) = 0 \]

This simplifies to:

\[ 9x - 10y + 12 = 0 \quad \text{(Equation 1)} \]

**For the second equation:**
- The LCM of 3 and 2 is also 6.
- Multiply the entire equation by 6:

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

This simplifies to:

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

### Step 2: Rearrange Equation 2 to express \(x\) in terms of \(y\)
From Equation 2:

\[ 2x + 3y = 2 \]

Rearranging gives:

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

### Step 3: Substitute \(x\) in Equation 1
Now, substitute \(x\) in Equation 1:

\[ 9\left(1 - \frac{3y}{2}\right) - 10y + 12 = 0 \]

Expanding this gives:

\[ 9 - \frac{27y}{2} - 10y + 12 = 0 \]

Combine like terms:

\[ 21 - \frac{27y}{2} - 10y = 0 \]

To combine the terms involving \(y\), convert \(10y\) to a fraction with a denominator of 2:

\[ 10y = \frac{20y}{2} \]

Now the equation becomes:

\[ 21 - \frac{27y + 20y}{2} = 0 \]
\[ 21 - \frac{47y}{2} = 0 \]

### Step 4: Solve for \(y\)
Rearranging gives:

\[ \frac{47y}{2} = 21 \]

Multiplying both sides by 2:

\[ 47y = 42 \]

Now, divide by 47:

\[ y = \frac{42}{47} \]

### Step 5: Substitute \(y\) back to find \(x\)
Now substitute \(y\) back into the equation for \(x\):

\[ x = 1 - \frac{3\left(\frac{42}{47}\right)}{2} \]

Calculating this gives:

\[ x = 1 - \frac{126}{94} \]
\[ x = \frac{94}{94} - \frac{126}{94} = -\frac{32}{94} = -\frac{16}{47} \]

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

\[ x = -\frac{16}{47}, \quad y = \frac{42}{47} \]