6. \( \left\{\begin{array}{l}\frac{1}{3} y-x=-2 \\ 3 x-y=7\end{array}\right. \)

To solve the system of equations: 1. Start with the first equation: \( \frac{1}{3}y - x = -2 \). Multiply through by 3 to eliminate the fraction: \[ y - 3x = -6 \quad \text{(Equation 1)} \] 2. Now take the second equation: \( 3x - y = 7 \). This can be rearranged to: \[ y = 3x - 7 \quad \text{(Equation 2)} \] 3. Now, substitute Equation 2 into Equation 1: \[ (3x - 7) - 3x = -6 \\ -7 = -6 \quad \text{(This is inconsistent)} \] This system of equations has no solution, indicating the lines represented by the equations are parallel. ### Fun Fact: Did you know that when you encounter systems of equations with no solutions like this one, it's because the lines are parallel? They never intersect, just like two friends who always seem to miss each other at the coffee shop! ### Real-World Connection: Parallel systems can also be seen in real life scenarios like two planes flying at the same altitude and heading in the same direction but never actually crossing paths. Understanding these concepts helps in fields ranging from engineering to physics, where trajectories and pathways need to be analyzed!