What is the slope of a line perpendicular to [tex]$y=-4 x+1$[/tex]?
Answer (1)
Identify the slope of the given line: The slope of y = − 4 x + 1 is − 4 .
Find the negative reciprocal: The negative reciprocal of − 4 is 4 1 .
State the slope of the perpendicular line: The slope of the line perpendicular to y = − 4 x + 1 is 4 1 .
Explanation
Understanding the Problem The problem asks us to find the slope of a line that is perpendicular to a given line. The given line is in slope-intercept form, which makes it easy to identify its slope.
Identifying the Slope of the Given Line The equation of the given line is y = − 4 x + 1 . The slope of this line is the coefficient of x , which is − 4 .
Understanding Perpendicular Lines Recall that two lines are perpendicular if and only if the product of their slopes is − 1 . This means that the slopes of perpendicular lines are negative reciprocals of each other.
Finding the Negative Reciprocal To find the slope of a line perpendicular to the given line, we need to find the negative reciprocal of − 4 . The negative reciprocal of a number is found by taking the reciprocal of the number and changing its sign.
Determining the Slope of the Perpendicular Line The reciprocal of − 4 is − 4 1 . Changing the sign gives us − ( − 4 1 ) = 4 1 . Therefore, the slope of the line perpendicular to y = − 4 x + 1 is 4 1 .
Examples
Understanding perpendicular slopes is crucial in various real-world applications, such as architecture and navigation. For example, when designing buildings, architects use perpendicular lines to ensure walls are at right angles, providing structural stability. In navigation, understanding perpendicular relationships helps in plotting courses and determining directions, ensuring accurate and safe travel.
