Select the correct answer.
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: y = − 4 1 x + 4 3 , so m 1 = − 4 1 .
Rewrite the second equation in slope-intercept form: y = 4 1 x − 2 1 , so m 2 = 4 1 .
Compare the slopes: m 1 = m 2 , so the lines are not parallel. Also, m 1 ⋅ m 2 = − 16 1 = − 1 , so the lines are not perpendicular.
Conclude that the lines are neither parallel nor perpendicular.
Explanation
Problem Analysis We are given two linear equations: 3 x + 12 y = 9 and 2 x − 8 y = 4 . Our objective is to determine whether these lines are parallel, perpendicular, or neither. To do this, we need to find the slopes of both lines and compare them.
Finding the Slope of the First Line First, let's rewrite the first equation, 3 x + 12 y = 9 , in slope-intercept form ( y = m x + b ), where m is the slope and b is the y-intercept. Subtract 3 x from both sides: 12 y = − 3 x + 9 . Now, divide both sides by 12: y = 12 − 3 x + 12 9 . Simplify the fractions: y = − 4 1 x + 4 3 . Thus, the slope of the first line is m 1 = − 4 1 .
Finding the Slope of the Second Line Next, let's rewrite the second equation, 2 x − 8 y = 4 , in slope-intercept form. Subtract 2 x from both sides: − 8 y = − 2 x + 4 . Now, divide both sides by -8: y = − 8 − 2 x + − 8 4 . Simplify the fractions: y = 4 1 x − 2 1 . Thus, the slope of the second line is m 2 = 4 1 .
Comparing the Slopes Now, we compare the slopes m 1 and m 2 . Parallel lines have the same slope, and perpendicular lines have slopes that are negative reciprocals of each other (their product is -1). In this case, m 1 = − 4 1 and m 2 = 4 1 . Since m 1 = m 2 , the lines are not parallel.
Checking for Perpendicularity To check if the lines are perpendicular, we multiply the slopes: m 1 ⋅ m 2 = ( − 4 1 ) ( 4 1 ) = − 16 1 . Since the product of the slopes is not -1, the lines are not perpendicular.
Conclusion Since the lines are neither parallel nor perpendicular, the correct answer is: The slopes of the lines are opposites, so they are neither parallel nor perpendicular.
Examples
Understanding whether lines are parallel or perpendicular is crucial in architecture and construction. For example, when designing a building, architects need to ensure that walls are perpendicular to the ground for stability. Similarly, parallel lines are used in road construction to maintain a constant distance between lanes, ensuring safe traffic flow. These concepts are also used in urban planning to design street grids and layouts.
The slopes of the lines obtained from the equations are m 1 = − 4 1 and m 2 = 4 1 . Since the slopes are different and their product is not -1, the correct option is that the lines are neither parallel nor perpendicular. Thus, the correct answer is option C: The slopes of the lines are opposites, so they are neither parallel nor perpendicular.
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