The graph of which of the following equations is a line perpendicular to the graph [tex]4x - 7y = 17[/tex]?
A. [tex]y = -\frac{7}{4}x + \frac{17}{4}[/tex]
B. [tex]y = \frac{7}{4}x + \frac{17}{4}[/tex]
C. [tex]y = \frac{4}{7}x + 17[/tex]
D. [tex]y = \frac{4}{7}x - \frac{17}{4}[/tex]
Answer (1)
Convert the given equation to slope-intercept form to find its slope: y = 7 4 x − 7 17 .
Determine the slope of the perpendicular line by taking the negative reciprocal of the original slope: − 4 7 .
Compare the slopes of the given options to the perpendicular slope.
The equation with the slope − 4 7 is the perpendicular line: y = − 4 7 x + 4 17 .
Explanation
Problem Analysis We are given the equation of a line 4 x − 7 y = 17 and asked to find which of the given options represents a line perpendicular to it. To solve this, we first need to determine the slope of the given line. Then, we find the negative reciprocal of that slope, which will be the slope of any line perpendicular to it. Finally, we compare this perpendicular slope to the slopes of the lines in the given options to find the correct answer.
Finding the Slope of the Given Line First, let's rewrite the given equation 4 x − 7 y = 17 in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. To do this, we isolate y :
4 x − 7 y = 17 − 7 y = − 4 x + 17 y = − 7 − 4 x + − 7 17 y = 7 4 x − 7 17
Finding the Slope of the Perpendicular Line From the slope-intercept form, we can see that the slope of the given line is m 1 = 7 4 . The slope of a line perpendicular to this line, m 2 , is the negative reciprocal of m 1 . Therefore, m 2 = − m 1 1 = − 7 4 1 = − 4 7 .
Comparing Slopes with the Options Now, we need to find which of the given options has a slope of − 4 7 . Let's examine each option:
a. y = − 4 7 x + 4 17 : This line has a slope of − 4 7 , which matches the slope of the perpendicular line we calculated. b. y = 4 7 x + 4 17 : This line has a slope of 4 7 .
c. y = 7 4 x + 17 : This line has a slope of 7 4 .
d. y = 7 4 x − 4 17 : This line has a slope of 7 4 .
Final Answer Option a, y = − 4 7 x + 4 17 , has a slope of − 4 7 , which is the negative reciprocal of the slope of the given line. Therefore, this line is perpendicular to the given line.
Conclusion The equation of the line perpendicular to 4 x − 7 y = 17 is y = − 4 7 x + 4 17 .
Examples
Understanding perpendicular lines is crucial in many real-world applications, such as architecture and construction. For example, when designing a building, architects need to ensure that walls are perpendicular to the ground to maintain structural integrity. Similarly, in navigation, understanding perpendicular relationships helps determine the shortest distance between two points or to calculate angles for course correction. These principles ensure safety and efficiency in various engineering and design projects.
