Solve : `{:((34)/(3x+4y)+(15)/(3x-2y)=5),((25)/(3x-2y)-(8.50)/(3x+4y)=4.5):}`

Answer

To solve the simultaneous linear equations given in the question, we will follow these steps:

### Step 1: Rewrite the equations
The given equations are:
1. \(\frac{34}{3x + 4y} + \frac{15}{3x - 2y} = 5\)
2. \(\frac{25}{3x - 2y} - \frac{8.5}{3x + 4y} = 4.5\)

### Step 2: Substitute variables
Let's make the equations easier to handle by substituting:
- Let \( a = \frac{1}{3x + 4y} \)
- Let \( b = \frac{1}{3x - 2y} \)

Now we can rewrite the equations as:
1. \( 34a + 15b = 5 \) (Equation 1)
2. \( 25b - 8.5a = 4.5 \) (Equation 2)

### Step 3: Rearrange Equation 2
We can rearrange Equation 2 to make it more manageable:
\[ -8.5a + 25b = 4.5 \]

### Step 4: Multiply Equation 2 by 4
To eliminate decimals, we can multiply the entire Equation 2 by 4:
\[ -34a + 100b = 18 \]

### Step 5: Set up the system of equations
Now we have the following system of equations:
1. \( 34a + 15b = 5 \)
2. \( -34a + 100b = 18 \)

### Step 6: Add the equations
Adding both equations will eliminate \( a \):
\[ (34a - 34a) + (15b + 100b) = 5 + 18 \]
This simplifies to:
\[ 115b = 23 \]

### Step 7: Solve for \( b \)
Now, divide both sides by 115:
\[ b = \frac{23}{115} = \frac{1}{5} \]

### Step 8: Substitute \( b \) back to find \( a \)
Now substitute \( b \) back into Equation 1 to find \( a \):
\[ 34a + 15 \left(\frac{1}{5}\right) = 5 \]
This simplifies to:
\[ 34a + 3 = 5 \]
Subtract 3 from both sides:
\[ 34a = 2 \]
Now divide by 34:
\[ a = \frac{2}{34} = \frac{1}{17} \]

### Step 9: Substitute back to find \( x \) and \( y \)
We have:
- \( a = \frac{1}{3x + 4y} \)
- \( b = \frac{1}{3x - 2y} \)

Substituting the values of \( a \) and \( b \):
1. From \( a = \frac{1}{17} \):
\[ 3x + 4y = 17 \quad \text{(Equation 3)} \]
2. From \( b = \frac{1}{5} \):
\[ 3x - 2y = 5 \quad \text{(Equation 4)} \]

### Step 10: Solve the system of equations (3 and 4)
Now we can solve Equations 3 and 4:
1. \( 3x + 4y = 17 \)
2. \( 3x - 2y = 5 \)

### Step 11: Subtract Equation 4 from Equation 3
Subtracting Equation 4 from Equation 3:
\[ (3x + 4y) - (3x - 2y) = 17 - 5 \]
This simplifies to:
\[ 6y = 12 \]
Dividing both sides by 6:
\[ y = 2 \]

### Step 12: Substitute \( y \) back to find \( x \)
Now substitute \( y = 2 \) back into Equation 3:
\[ 3x + 4(2) = 17 \]
This simplifies to:
\[ 3x + 8 = 17 \]
Subtracting 8 from both sides:
\[ 3x = 9 \]
Dividing both sides by 3:
\[ x = 3 \]

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