Determine whether the graphs of the equations are parallel, perpendicular, or neither. \[ \begin{array}{l} 2 y=4 x-8 \\ 4 y-9=2 x \end{array} \] Are the lines parallel, perpendicular, or neither? A. The lines are perpendicular. B. The lines are neither parallel nor perpendicular: C. The lines are parallel. ot started

To determine the relationship between the lines represented by the equations, we first need to put them into slope-intercept form (y = mx + b), where m represents the slope. 1. For the first equation: \[ 2y = 4x - 8 \\ y = 2x - 4 \] The slope (m) of the first line is 2. 2. For the second equation: \[ 4y - 9 = 2x \\ 4y = 2x + 9 \\ y = \frac{1}{2}x + \frac{9}{4} \] The slope (m) of the second line is \(\frac{1}{2}\). Now, we analyze the slopes: - The first line's slope is 2. - The second line's slope is \(\frac{1}{2}\). To determine if the lines are parallel, we check if their slopes are equal: - They are not equal, so the lines are not parallel. To see if they are perpendicular, we multiply their slopes: - \(2 \times \frac{1}{2} = 1\) Since the product of their slopes is 1, we conclude that the lines are indeed perpendicular. So, the answer is: **A. The lines are perpendicular.** --- Both equations have interesting historical contexts. The systematic study of linear equations dates back to ancient civilizations such as the Babylonians, who used geometric methods to solve problems that involved straight lines and proportions. They laid early foundations for algebra! On a practical note, understanding whether lines are parallel or perpendicular is essential in many fields such as architecture, engineering, and computer graphics. It helps in designing structures, creating 2D and 3D models, and even programming for video games, ensuring that lines interact correctly and form appropriate angles.