Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. \[ \begin{array}{r}-2 x+3 y=-1 \\ \qquad x-3 y=3\end{array} \] Answer Infinitely Many Solutions No Solutions One Solution

To determine the relationship between the two equations, we can use the method of elimination or substitution. By rearranging the second equation \(x - 3y = 3\) to express \(x\) in terms of \(y\), we have \(x = 3 + 3y\). Substituting this into the first equation gives us a single variable equation, helping us find a specific value for \(y\). Solving eventually leads to a consistent intersection of values for both equations, indicating there is typically exactly one solution. However, a more colorful approach reveals that if two lines represented by the equations have the same slope but different intercepts, they will never meet, thus leading to no solutions. Conversely, if they share the same slope and intercept, they overlap completely, meaning infinitely many solutions. Thus, solving these equations confirms that they intersect at just one point: therefore, the system has exactly one solution!