Use the point-slope formula to write an equation of the line that passes through $\left(\frac{3}{4},-\frac{4}{9}\right)$ and has an undefined slope. Write the answer in slope-intercept form (if possible).
Answer (1)
Recognize that an undefined slope indicates a vertical line.
Vertical lines have the form x = c .
Since the line passes through ( 4 3 , − 9 4 ) , the equation is x = 4 3 .
Vertical lines cannot be written in slope-intercept form, so the final answer is x = 4 3 .
Explanation
Understanding the Problem We are given a point ( 4 3 , − 9 4 ) and the fact that the line has an undefined slope. Our goal is to find the equation of this line and express it in slope-intercept form if possible.
Vertical Lines A line with an undefined slope is a vertical line. Vertical lines have the equation x = c , where c is a constant. This is because for all points on the line, the x -coordinate is the same, while the y -coordinate can vary.
Finding the Equation Since the line passes through the point ( 4 3 , − 9 4 ) , the x -coordinate of this point must be the constant c in the equation x = c . Therefore, the equation of the line is x = 4 3 .
Slope-Intercept Form The slope-intercept form of a line is y = m x + b , where m is the slope and b is the y -intercept. However, vertical lines have an undefined slope, so they cannot be written in slope-intercept form. The equation x = 4 3 represents a vertical line and cannot be rearranged into the form y = m x + b .
Final Answer Therefore, the equation of the line is x = 4 3 , and it cannot be written in slope-intercept form.
Examples
Imagine you're designing a fence that must be perfectly vertical. This problem helps you determine the exact placement of the fence posts. If one post is at a specific x-coordinate, say 4 3 meters from a reference point, and the fence must be vertical, then all other points on the fence must also have the same x-coordinate. This ensures the fence is straight and doesn't lean. Understanding vertical lines is crucial in construction and design for maintaining stability and accuracy.
