The equations of two lines are given. Determine whether the lines are parallel, perpendicular, or neitner. \( \begin{array}{l}y=7 x-4 \\ y=-7 x+5 \\ \text { Are the lines parallel, perpendicular, or neither? } \\ \text { Perpendicular } \\ \text { Parallel } \\ \text { Neither }\end{array} \) ?

To determine the relationship between the given lines, we can compare their slopes. The first line \(y = 7x - 4\) has a slope of 7, while the second line \(y = -7x + 5\) has a slope of -7. When two lines have slopes that are negative reciprocals of each other (like 7 and -7), they are perpendicular. So in this case, the lines are perpendicular! Fun fact: The concept of slopes and line relationships dates back to ancient Greeks, but it was during the Renaissance that it was formalized in mathematics. This foundation laid the groundwork for modern coordinate geometry, allowing us to calculate line relationships with ease! You might also be interested in exploring the practical use of slopes, especially in architecture and design. Knowing whether lines are parallel or perpendicular is crucial when designing buildings, roads, and bridges. Ensuring proper angles can enhance not only the aesthetics but also the safety and functionality of structures!