Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. \[ \begin{array}{l}-x+2 y=-7 \\ -4 x+8 y=-28\end{array} \] Answer Infinitely Many Solutions No Solutions One Solution

To analyze the system of equations, we'll first simplify the second equation: The second equation \(-4x + 8y = -28\) can be simplified by dividing every term by -4, giving us: \[ x - 2y = 7 \] Now, we can rewrite this in a standard form: \[ x - 2y = 7 \] Now we can compare this with the first equation which is already in standard form: \[ -x + 2y = -7 \] If we multiply the first equation by -1, we get: \[ x - 2y = 7 \] Both equations are essentially the same, meaning they represent the same line. Therefore, the system has infinitely many solutions, as any point on this line will satisfy both equations. For a visual twist, imagine these lines as two dancers spinning on stage, with each one following the other's steps perfectly. Since they never miss a beat, they end up dancing together for all time, illustrating the infinitely many solutions! As a little tip, when faced with such systems, always check if you can simplify the equations first, and remember that identical equations signal a dance of infinite possibilities! Just watch out; if the two equations were parallel but not the same, that would point to two dancers who can’t quite sync up—no solution there!