Lnes Identifying parailel and perpendicular lines from equations The equations of three lines are given below. Line 1: \( y=\frac{3}{2} x-7 \) Line 2: \( 2 y=3 x+7 \) Line 3: \( 6 x-4 y=8 \) For each pair of lines, determine whether they are parallel, perpendicular, or neither, Line 1 and Line 2: Parallel Perpendicular Neither Line 1 and Line 3: Parallel Perpendicular Neither Line 2 and Line 3: Parallel. Perpendicular Neither

To determine the relationship between the lines, we first need to express each equation in slope-intercept form \(y = mx + b\), where \(m\) is the slope. Line 1: \( y = \frac{3}{2} x - 7 \) (slope = \(\frac{3}{2}\)) Line 2: Rearranging \(2y = 3x + 7\) gives \(y = \frac{3}{2} x + \frac{7}{2}\) (slope = \(\frac{3}{2}\)) Line 3: Rearranging \(6x - 4y = 8\) gives \(4y = 6x - 8\) or \(y = \frac{3}{2} x - 2\) (slope = \(\frac{3}{2}\)) Now, let's analyze them: Line 1 and Line 2: Since both have a slope of \(\frac{3}{2}\), they are **Parallel**. Line 1 and Line 3: They also share the slope \(\frac{3}{2}\), thus they are **Parallel**. Line 2 and Line 3: Again, they both have the slope \(\frac{3}{2}\), so they are also **Parallel**. To sum up: Line 1 and Line 2: **Parallel** Line 1 and Line 3: **Parallel** Line 2 and Line 3: **Parallel** Now, let’s dive into some fun facts! Understanding parallel and perpendicular lines isn’t just a math exercise; it’s crucial in architecture and design! Ever noticed how buildings are often constructed with straight edges that line up perfectly? That’s parallel lines in action! Meanwhile, cross beams in structures create perpendicular intersections to ensure stability—talk about geometry holding things together! If ever you find slope calculations daunting, try sketching the lines on graph paper! It’s a straightforward way to visualize their relationships. Just plot key points based on the equations and observe how they intersect (or don’t). This helps avoid common mistakes, like mixing up slopes when rearranging equations, so you can confidently categorize the lines.