The equations of three lines are given below.
Line 1: [tex]$y=\frac{4}{3} x-4$[/tex]
Line 2: [tex]$6 x+8 y=4$[/tex]
Line 3: [tex]$3 y=4 x+2$[/tex]
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
Answer (2)
Find the slopes of the lines: Rewrite each equation in the form y = m x + b to identify the slope m .
Compare the slopes: Lines are parallel if their slopes are equal, and perpendicular if the product of their slopes is -1.
Line 1 and Line 2: Slopes are 3 4 and − 4 3 . Since 3 4 × − 4 3 = − 1 , they are perpendicular.
Line 1 and Line 3: Slopes are both 3 4 , so they are parallel.
Line 2 and Line 3: Slopes are − 4 3 and 3 4 . Since − 4 3 × 3 4 = − 1 , they are perpendicular. Line 1 and Line 2: Perpendicular, Line 1 and Line 3: Parallel, Line 2 and Line 3: Perpendicular
Explanation
Understanding the Problem We are given three lines and we need to determine whether each pair of lines are parallel, perpendicular, or neither. To do this, we will first find the slope of each line by rewriting the equations in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. Then, we will compare the slopes of each pair of lines. If the slopes are equal, the lines are parallel. If the product of the slopes is -1, the lines are perpendicular. Otherwise, the lines are neither parallel nor perpendicular.
Finding the Slopes Let's find the slope of each line.
Line 1: y = 3 4 x − 4 . The slope of Line 1 is m 1 = 3 4 .
Line 2: 6 x + 8 y = 4 . We need to rewrite this equation in slope-intercept form. Subtract 6 x from both sides: 8 y = − 6 x + 4 . Divide both sides by 8: y = − 8 6 x + 8 4 . Simplify the fractions: y = − 4 3 x + 2 1 . The slope of Line 2 is m 2 = − 4 3 .
Line 3: 3 y = 4 x + 2 . Divide both sides by 3: y = 3 4 x + 3 2 . The slope of Line 3 is m 3 = 3 4 .
Comparing the Slopes Now, let's compare the slopes of each pair of lines.
Line 1 and Line 2: m 1 = 3 4 and m 2 = − 4 3 . The product of the slopes is m 1 ⋅ m 2 = 3 4 ⋅ − 4 3 = − 1 . Since the product of the slopes is -1, Line 1 and Line 2 are perpendicular.
Line 1 and Line 3: m 1 = 3 4 and m 3 = 3 4 . Since the slopes are equal, Line 1 and Line 3 are parallel.
Line 2 and Line 3: m 2 = − 4 3 and m 3 = 3 4 . The product of the slopes is m 2 ⋅ m 3 = − 4 3 ⋅ 3 4 = − 1 . Since the product of the slopes is -1, Line 2 and Line 3 are perpendicular.
Final Answer Therefore, we have:
Line 1 and Line 2: Perpendicular Line 1 and Line 3: Parallel Line 2 and Line 3: Perpendicular
Examples
Understanding the relationships between lines (parallel, perpendicular, or neither) is crucial in various real-world applications. For instance, architects and engineers use these concepts to design buildings and structures, ensuring walls are perpendicular for stability and floors are parallel for even weight distribution. City planners use parallel lines for roads and perpendicular lines for intersections to optimize traffic flow and minimize accidents. Even in art and design, understanding these relationships helps create visually appealing and balanced compositions.
Lines 1 and 2 are perpendicular, Lines 1 and 3 are parallel, and Lines 2 and 3 are perpendicular. This is determined by calculating the slopes of the lines and comparing them. If the slopes are equal, the lines are parallel; if the product of the slopes is -1, the lines are perpendicular.
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