Solve the following paris of linear (simultaneous) equation using method of elimination by substution:
`(2x + 1)/(7) + (5y - 3)/(3) = 12`
`(3x + 2)/(2) - (4y - 3)/(9) = 13`

Answer

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

### Step 1: Simplify the First Equation
The first equation is:
\[ \frac{2x + 1}{7} + \frac{5y - 3}{3} = 12 \]
To eliminate the fractions, we will find the least common multiple (LCM) of the denominators, which is 21. We will multiply the entire equation by 21:
\[ 21 \left(\frac{2x + 1}{7}\right) + 21 \left(\frac{5y - 3}{3}\right) = 21 \cdot 12 \]
This simplifies to:
\[ 3(2x + 1) + 7(5y - 3) = 252 \]
Expanding this gives:
\[ 6x + 3 + 35y - 21 = 252 \]
Combining like terms:
\[ 6x + 35y - 18 = 252 \]
Now, adding 18 to both sides:
\[ 6x + 35y = 270 \]
Let’s mark this as Equation 1:
\[ \text{Equation 1: } 6x + 35y = 270 \]

### Step 2: Simplify the Second Equation
The second equation is:
\[ \frac{3x + 2}{2} - \frac{4y - 3}{9} = 13 \]
Again, we will find the LCM of the denominators, which is 18. We will multiply the entire equation by 18:
\[ 18 \left(\frac{3x + 2}{2}\right) - 18 \left(\frac{4y - 3}{9}\right) = 18 \cdot 13 \]
This simplifies to:
\[ 9(3x + 2) - 2(4y - 3) = 234 \]
Expanding this gives:
\[ 27x + 18 - 8y + 6 = 234 \]
Combining like terms:
\[ 27x - 8y + 24 = 234 \]
Now, subtracting 24 from both sides:
\[ 27x - 8y = 210 \]
Let’s mark this as Equation 2:
\[ \text{Equation 2: } 27x - 8y = 210 \]

### Step 3: Solve for One Variable
We will solve Equation 2 for \(x\):
\[ 27x = 210 + 8y \]
Dividing both sides by 27 gives:
\[ x = \frac{210 + 8y}{27} \]

### Step 4: Substitute into the First Equation
Now, we will substitute this expression for \(x\) into Equation 1:
\[ 6\left(\frac{210 + 8y}{27}\right) + 35y = 270 \]
Multiplying through by 27 to eliminate the fraction:
\[ 6(210 + 8y) + 35y \cdot 27 = 270 \cdot 27 \]
This simplifies to:
\[ 1260 + 48y + 945y = 7290 \]
Combining like terms:
\[ 1260 + 993y = 7290 \]
Subtracting 1260 from both sides:
\[ 993y = 7290 - 1260 \]
Calculating the right side:
\[ 993y = 6030 \]
Dividing both sides by 993 gives:
\[ y = \frac{6030}{993} = 6 \]

### Step 5: Substitute \(y\) Back to Find \(x\)
Now that we have \(y\), we substitute it back into the expression for \(x\):
\[ x = \frac{210 + 8(6)}{27} \]
Calculating this gives:
\[ x = \frac{210 + 48}{27} = \frac{258}{27} = 10 \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = 10, \quad y = 6 \]