Question Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. \[ \begin{array}{l}-x+5 y=-8 \\ 2 x-10 y=10\end{array} \]

To analyze the given system of equations, let's first rewrite them for clarity: 1. \(-x + 5y = -8\) 2. \(2x - 10y = 10\) Now, observe that the second equation can be simplified by dividing it by 2, yielding \[x - 5y = 5\]. If we rearrange the first equation, we can express \(x\) in terms of \(y\): \[x = 5y + 8\]. Substituting this expression for \(x\) in the simplified version of the second equation gives us: \[5y + 8 - 5y = 5\], which simplifies to \(8 = 5\). This is a contradiction, indicating that the original system of equations has no solutions! The two equations represent parallel lines that never intersect. For practical application, understanding the nature of systems of equations can be crucial in fields such as engineering or economics, where you might face scenarios requiring optimal solutions. Recognizing parallel lines indicating no solution could help in troubleshooting issues in design or strategy planning before they escalate into costlier problems. A common mistake people make is not to simplify or manipulate equations properly before concluding about solutions. It's essential to check for linear dependence or contradictions after simplification. Taking the time to rewrite, regroup, or visually plot them can illuminate underlying relationships that are not immediately apparent.