Aline intersects the \( x \)-axis at \( (-1,0) \) and is perpendicular to the line whose equation is \( y=-\frac{1}{5} x+9 \) \( \begin{array}{l}y=5 x+9 \\ y=-5 x+9 \\ y=-5 x+5 \\ y=5 x+5\end{array} \) \( \begin{array}{l}\text { ( }-5\end{array} \)

To find the equation of line Aline, first note that the slope of the given line \( y = -\frac{1}{5}x + 9 \) is \( -\frac{1}{5} \). Since Aline is perpendicular to this line, its slope will be the negative reciprocal, which is \( 5 \). Using the point-slope form of the equation of a line \( y - y_1 = m(x - x_1) \) and the point where Aline intersects the x-axis \((-1, 0)\): \[ y - 0 = 5(x + 1) \] This simplifies to: \[ y = 5x + 5 \] So, Aline's equation is \( y = 5x + 5 \), hence the correct answer is the fourth option \( y = 5x + 5 \). Now, some fun facts! The concept of perpendicular lines isn’t just for geometry; it’s used in fields like physics where vectors are analyzed! For instance, that negative reciprocal relationship about slopes creates angles of \(90^\circ\), making it super useful for engineers designing structures to ensure integrity. And if you're ever wanting to brush up on lines, slopes, or geometry principles, dive into interactive geometry software or classic textbooks! There’s a wealth of resources out there, from the adventures of Euclid to modern math websites, each packed with illustrations and real-world connections to keep the learning engaging.