Solve for x and y :
`(y+7)/(5)=(2y-x)/(4)+3x-5`
`(7-5x)/(2)+(3-4y)/(6)=5y-18`

Answer

To solve the given simultaneous linear equations for \(x\) and \(y\):

1. **Write down the equations:**
\[ \frac{y + 7}{5} = \frac{2y - x}{4} + 3x - 5 \quad \text{(Equation 1)} \]
\[ \frac{7 - 5x}{2} + \frac{3 - 4y}{6} = 5y - 18 \quad \text{(Equation 2)} \]

2. **Clear the fractions in Equation 1:**
Multiply through by 20 (the least common multiple of the denominators 5 and 4):
\[ 20 \cdot \frac{y + 7}{5} = 20 \cdot \left( \frac{2y - x}{4} + 3x - 5 \right) \]
This simplifies to:
\[ 4(y + 7) = 5(2y - x) + 60x - 100 \]
Expanding both sides:
\[ 4y + 28 = 10y - 5x + 60x - 100 \]
Combine like terms:
\[ 4y + 28 = 10y + 55x - 100 \]
Rearranging gives:
\[ 55x + 6y = 128 \quad \text{(Equation 1 simplified)} \]

3. **Clear the fractions in Equation 2:**
Multiply through by 6 (the least common multiple of the denominators 2 and 6):
\[ 6 \cdot \left( \frac{7 - 5x}{2} + \frac{3 - 4y}{6} \right) = 6(5y - 18) \]
This simplifies to:
\[ 3(7 - 5x) + (3 - 4y) = 30y - 108 \]
Expanding both sides:
\[ 21 - 15x + 3 - 4y = 30y - 108 \]
Combine like terms:
\[ 24 - 15x - 4y = 30y - 108 \]
Rearranging gives:
\[ 15x + 34y = 132 \quad \text{(Equation 2 simplified)} \]

4. **Now we have a system of equations:**
\[ 55x + 6y = 128 \quad \text{(1)} \]
\[ 15x + 34y = 132 \quad \text{(2)} \]

5. **Solve for \(y\) in terms of \(x\) from Equation (1):**
\[ 6y = 128 - 55x \]
\[ y = \frac{128 - 55x}{6} \]

6. **Substitute \(y\) into Equation (2):**
\[ 15x + 34\left(\frac{128 - 55x}{6}\right) = 132 \]
Multiply through by 6 to eliminate the fraction:
\[ 90x + 34(128 - 55x) = 792 \]
Expanding gives:
\[ 90x + 4352 - 1870x = 792 \]
Combine like terms:
\[ -1780x + 4352 = 792 \]
Rearranging gives:
\[ -1780x = 792 - 4352 \]
\[ -1780x = -3560 \]
Dividing by -1780:
\[ x = 2 \]

7. **Substitute \(x\) back into the equation for \(y\):**
\[ y = \frac{128 - 55(2)}{6} \]
\[ y = \frac{128 - 110}{6} \]
\[ y = \frac{18}{6} = 3 \]

8. **Final solution:**
\[ x = 2, \quad y = 3 \]