Question Find the slope of a line perpendicular to the line whose equation is \( 5 x+6 y=-18 \). Fully simplify your answer.

To find the slope of a line perpendicular to the line given by the equation \( 5x + 6y = -18 \), we first need to convert it into slope-intercept form \( y = mx + b \). Rearranging the equation, we isolate \( y \): \[ 6y = -5x - 18 \quad \Rightarrow \quad y = -\frac{5}{6}x - 3 \] From this, we see that the slope (m) of the original line is \( -\frac{5}{6} \). The slope of a line perpendicular to this line is the negative reciprocal. Therefore, we take the reciprocal of \( -\frac{5}{6} \) and change the sign: \[ \text{Perpendicular slope} = \frac{6}{5} \] This gives us the final answer, fully simplified as \( \frac{6}{5} \). Now let’s add some fun facts! If you ever wondered why slopes matter, think of them as the steepness of hills! When you're hiking, a slope of \( \frac{6}{5} \) means for every 5 units you move horizontally, you climb 6 units vertically. Did you know that perpendicular lines are all about relationships? In geometry, when two lines intersect at a right angle, they provide a perfect basis for creating rectangles and squares, the fundamental building blocks of architecture!