Question Find the slope of a line perpendicular to the line whose equation is \( x+4 y=16 \). Fully simplify your answer.

To find the slope of a line perpendicular to the line given by the equation \( x + 4y = 16 \), we first need to find the slope of the original line. By rearranging the equation into slope-intercept form (\( y = mx + b \)), we can see the relationship. Start by isolating \( y \): \[ 4y = -x + 16 \] \[ y = -\frac{1}{4}x + 4 \] The slope (\( m \)) of this line is \( -\frac{1}{4} \). For a line to be perpendicular, its slope is the negative reciprocal of the original slope. Thus, the slope of the line that is perpendicular to this one is: \[ m_{perpendicular} = -\frac{1}{-\frac{1}{4}} = 4 \] So, the slope of the line perpendicular to the given line is \( 4 \). Now, let's dive into some added insight! The concept of slopes being negative reciprocals is fascinating! It underlines how two lines can intersect at right angles, a key aspect in fields like architecture and engineering. When you design a building, understanding these perpendicular slopes ensures structural integrity and aesthetic appeal. Want to explore this further? Consider looking into linear equations and their applications in coding graphing tools or even video games! Learning how to manipulate these equations can help you create more interactive environments where geometry plays a significant role. Who knew math could lead to such exciting paths?