Question Find the slope of a line perpendicular to the line whose equation is \( 3 x-6 y=90 \). Fully simplify your answer.

To find the slope of a line perpendicular to the given line, we first need to determine the slope of the original line. We can rewrite the equation \( 3x - 6y = 90 \) in slope-intercept form \( y = mx + b \). Rearranging, we get: \[ -6y = -3x + 90 \] Dividing by -6 yields: \[ y = \frac{1}{2}x - 15 \] The slope (\( m \)) of the given line is \( \frac{1}{2} \). The slope of a line perpendicular to another is the negative reciprocal of the original slope. Thus, the slope we seek is: \[-\frac{1}{\frac{1}{2}} = -2\] So, the slope of a line perpendicular to the given line is \( -2 \). Now, here’s some bonus info! You may not realize this, but the concept of slope dates back to ancient mathematicians working on geometry. The Greeks, such as Euclid, studied angles and lines, laying the groundwork for how we understand linear relationships today! It’s fascinating how these early ideas still influence our modern math—as you’re discovering through algebra! On a practical note, knowing how to calculate and apply slopes is super useful in various real-world settings. Engineers utilize slopes when designing roads and ramps, ensuring that they are safely inclined to facilitate driving and accessibility. This math isn't just academic; it literally shapes the world we move in every day!