Are the given lines parallel, perpendicular, or neither?
[tex]
\begin{array}{l}
3 x+12 y=9 \\
2 x-8 y=4
\end{array}
[/tex]
A. The quotient of the slopes of the lines is 1, so the lines are parallel.
B. The slopes of the lines are not the same, so they are perpendicular.
C. The slopes of the lines are opposites, so they are neither parallel nor perpendicular.
D. The product of the slopes of the lines is 1, so the lines are perpendicular.
Answer (2)
Rewrite the first equation in slope-intercept form to find its slope: y = − 4 1 x + 4 3 , so m 1 = − 4 1 .
Rewrite the second equation in slope-intercept form to find its slope: y = 4 1 x − 2 1 , so m 2 = 4 1 .
Check if the slopes are equal (parallel): m 1 = m 2 , so the lines are not parallel.
Check if the product of the slopes is -1 (perpendicular): m 1 ⋅ m 2 = − 16 1 = − 1 , so the lines are not perpendicular.
Conclude that the lines are neither parallel nor perpendicular: $\boxed{\text{neither}}.
Explanation
Problem Analysis We are given two linear equations: 3 x + 12 y = 9 and 2 x − 8 y = 4 . Our goal is to determine whether these lines are parallel, perpendicular, or neither. To do this, we will rewrite each equation in slope-intercept form, which is y = m x + b , where m represents the slope of the line.
Finding the Slope of the First Line First, let's rewrite the first equation, 3 x + 12 y = 9 , in slope-intercept form. We need to isolate y :
Subtract 3 x from both sides: 12 y = − 3 x + 9
Divide both sides by 12 : y = 12 − 3 x + 12 9
Simplify the fractions: y = − 4 1 x + 4 3
So, the slope of the first line, m 1 , is − 4 1 .
Finding the Slope of the Second Line Now, let's rewrite the second equation, 2 x − 8 y = 4 , in slope-intercept form. We need to isolate y :
Subtract 2 x from both sides: − 8 y = − 2 x + 4
Divide both sides by − 8 : y = − 8 − 2 x + − 8 4
Simplify the fractions: y = 4 1 x − 2 1
So, the slope of the second line, m 2 , is 4 1 .
Determining the Relationship Between the Lines Now that we have the slopes of both lines, m 1 = − 4 1 and m 2 = 4 1 , we can determine the relationship between the lines.
Parallel Lines: Parallel lines have the same slope. Since m 1 = m 2 , the lines are not parallel.
Perpendicular Lines: Perpendicular lines have slopes that are negative reciprocals of each other. In other words, the product of their slopes is − 1 . Let's check if m 1 ⋅ m 2 = − 1 :
m 1 ⋅ m 2 = ( − 4 1 ) ⋅ ( 4 1 ) = − 16 1 Since − 16 1 = − 1 , the lines are not perpendicular.
Since the lines are neither parallel nor perpendicular, the correct answer is that the lines are neither parallel nor perpendicular.
Final Answer The slopes of the lines are m 1 = − 4 1 and m 2 = 4 1 . Since the slopes are not the same and their product is not -1, the lines are neither parallel nor perpendicular.
Examples
Understanding whether lines are parallel, perpendicular, or neither is crucial in various real-world applications. For instance, architects use these concepts to design buildings, ensuring walls are perpendicular for stability or creating parallel lines for aesthetic appeal. City planners use these principles to design road layouts, optimizing traffic flow and minimizing accidents. In computer graphics, determining the relationship between lines is essential for rendering images and creating realistic visual effects. These concepts are fundamental in fields requiring precision and spatial reasoning.
The slopes of the lines are m 1 = − 4 1 and m 2 = 4 1 . Since the slopes are not the same and their product is not -1, the lines are neither parallel nor perpendicular. Therefore, the correct answer is that the lines are neither parallel nor perpendicular.
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