Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. \[ \begin{array}{l}-2 x+y=-3 \\ 6 x-3 y=7\end{array} \] Answer No Solutions Infinitely Many Solutions Submit Answre One Solution

To determine the nature of the solutions for the system of equations given, we can express one equation in terms of the other. The first equation can be rearranged to \( y = 2x - 3 \). Substituting this \( y \) into the second equation \( 6x - 3(2x - 3) = 7 \) simplifies to \( 6x - 6x + 9 = 7 \), resulting in \( 9 = 7 \), which is a contradiction. Since we arrive at a contradiction, this system of equations has no solutions. For a fun twist, did you know that when a system of equations has no solutions, such lines are called "parallel lines"? They run side by side without ever crossing, just like two friends who can never agree on the same restaurant!