`(1)/(3)x-(1)/(6)y=7`
`(1)/(5)y-(1)/(5)x=8`
Which of the following ordered pairs (x, y) satisfies the system of equations above?

Options

A. `(-36, -57)`
B. `(12, 43)`
C. `((101)/(5), (307)/(4))`
D. `(82, 122)`

Answer

To solve the system of equations:

1. **Equations Given**:
\[ \frac{1}{3}x - \frac{1}{6}y = 7 \quad (1) \]
\[ \frac{1}{5}y - \frac{1}{5}x = 8 \quad (2) \]

2. **Eliminate Fractions in Equation (1)**:
- The least common multiple (LCM) of the denominators (3 and 6) is 6.
- Multiply the entire equation by 6:
\[ 6 \left(\frac{1}{3}x\right) - 6 \left(\frac{1}{6}y\right) = 6 \cdot 7 \]
- This simplifies to:
\[ 2x - y = 42 \quad (3) \]

3. **Eliminate Fractions in Equation (2)**:
- The LCM of the denominators (5 and 5) is 5.
- Multiply the entire equation by 5:
\[ 5 \left(\frac{1}{5}y\right) - 5 \left(\frac{1}{5}x\right) = 5 \cdot 8 \]
- This simplifies to:
\[ y - x = 40 \quad (4) \]

4. **Rearranging Equation (4)**:
- Rearranging gives:
\[ y = x + 40 \quad (5) \]

5. **Substituting Equation (5) into Equation (3)**:
- Substitute \(y\) from (5) into (3):
\[ 2x - (x + 40) = 42 \]
- Simplifying this gives:
\[ 2x - x - 40 = 42 \]
\[ x - 40 = 42 \]
\[ x = 42 + 40 = 82 \quad (6) \]

6. **Finding \(y\)**:
- Substitute \(x\) back into (5):
\[ y = 82 + 40 = 122 \quad (7) \]

7. **Final Ordered Pair**:
- The solution to the system of equations is:
\[ (x, y) = (82, 122) \]

8. **Conclusion**:
- The ordered pair that satisfies the system of equations is \( (82, 122) \), which corresponds to option D.