Consider the line [tex]y=\frac{2}{3} x-4[/tex].
A line parallel to the graph of the line would have a slope of $\square$ .
A line perpendicular to the graph of the line would have a slope of $\square$.
Answer (1)
The slope of the given line y = 3 2 x − 4 is 3 2 .
Parallel lines have the same slope, so the slope of a line parallel to the given line is 3 2 .
Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of 3 2 is − 2 3 .
The slope of a line parallel to the given line is 3 2 and the slope of a line perpendicular to the given line is − 2 3 .
Explanation
Understanding the Problem The given line is y = 3 2 x − 4 . We need to find the slope of a line parallel to this line and the slope of a line perpendicular to it.
Identifying the Slope The slope of the given line is the coefficient of x , which is 3 2 .
Parallel Line Slope A line parallel to the given line will have the same slope. Therefore, the slope of a line parallel to y = 3 2 x − 4 is 3 2 .
Perpendicular Line Slope A line perpendicular to the given line will have a slope that is the negative reciprocal of the original slope. The negative reciprocal of 3 2 is − 2 3 .
Final Answer Therefore, a line parallel to the given line has a slope of 3 2 , and a line perpendicular to the given line has a slope of − 2 3 .
Examples
Understanding parallel and perpendicular slopes is crucial in various real-world applications, such as architecture and navigation. For instance, when designing buildings, architects ensure that walls are perpendicular to the ground for stability. Similarly, in navigation, understanding perpendicular and parallel paths helps in plotting efficient routes. Knowing that parallel lines have the same slope and perpendicular lines have negative reciprocal slopes allows for precise calculations and designs in these fields.
