Solve the following using method of by substitution
`(3x)/(2)-(5y)/(3)+2=0`
`(x)/(3)+(y)/(2)=2.(1)/(6)`

Answer

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

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

### Step 2: Simplify the first equation
We will first simplify the first equation. To eliminate the fractions, we can find the least common multiple (LCM) of the denominators (2 and 3), which is 6.

Multiplying the entire equation by 6:
\[ 6 \left(\frac{3x}{2}\right) - 6 \left(\frac{5y}{3}\right) + 6(2) = 0 \]
This simplifies to:
\[ 9x - 10y + 12 = 0 \]
Rearranging gives:
\[ 9x - 10y = -12 \quad \text{(Equation 1)} \]

### Step 3: Simplify the second equation
Now, let's simplify the second equation. First, convert \(2 \frac{1}{6}\) to an improper fraction:
\[ 2 \frac{1}{6} = \frac{13}{6} \]
Now, we rewrite the equation:
\[ \frac{x}{3} + \frac{y}{2} = \frac{13}{6} \]
Multiplying through by 6 to eliminate the fractions:
\[ 6 \left(\frac{x}{3}\right) + 6 \left(\frac{y}{2}\right) = 13 \]
This simplifies to:
\[ 2x + 3y = 13 \quad \text{(Equation 2)} \]

### Step 4: Solve for one variable
From Equation 1, we can express \(x\) in terms of \(y\):
\[ 9x = 10y - 12 \]
\[ x = \frac{10y - 12}{9} \quad \text{(Equation 3)} \]

### Step 5: Substitute into the second equation
Now, substitute Equation 3 into Equation 2:
\[ 2\left(\frac{10y - 12}{9}\right) + 3y = 13 \]
Multiplying through by 9 to eliminate the fraction:
\[ 2(10y - 12) + 27y = 117 \]
Expanding gives:
\[ 20y - 24 + 27y = 117 \]
Combining like terms:
\[ 47y - 24 = 117 \]
Adding 24 to both sides:
\[ 47y = 141 \]
Dividing by 47:
\[ y = 3 \]

### Step 6: Substitute back to find x
Now that we have \(y\), substitute \(y = 3\) back into Equation 3 to find \(x\):
\[ x = \frac{10(3) - 12}{9} \]
Calculating gives:
\[ x = \frac{30 - 12}{9} = \frac{18}{9} = 2 \]

### Final Answer
The solution to the system of equations is:
\[ x = 2, \quad y = 3 \]