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

The system of equations given can be analyzed by manipulating the equations into a more comparable form. If we add the two equations together, we get \( (2x - 2x) + (y - y) = 7 - 7 \), which simplifies to \( 0 = 0 \). This indicates that the equations are actually dependent; one can be derived from the other. Thus, it points to the fact that there may be infinitely many solutions rather than just one or none. Upon further inspection, if you solve one of the equations for \( y \), let's say from \( 2x + y = 7 \), you get \( y = 7 - 2x \). Substituting this into the second equation will yield that both equations describe the same line, confirming the presence of infinitely many solutions that satisfy \( 2x + y = 7 \) for all points on that line!