Find the equation of the line that is perpendicular to $y=-4 x+3$ and contains the point $(8,1)$.
$y=\frac{[?]}{[]} x+[]$
Answer (1)
Determine the slope of the given line: m 1 = − 4 .
Calculate the slope of the perpendicular line: m 2 = 4 1 .
Use the point-slope form with the point ( 8 , 1 ) and the perpendicular slope: y − 1 = 4 1 ( x − 8 ) .
Convert to slope-intercept form: y = 4 1 x − 1 . The final answer is y = 4 1 x − 1 .
Explanation
Understanding the Problem We are given a line y = − 4 x + 3 and a point ( 8 , 1 ) . We need to find the equation of a line that is perpendicular to the given line and passes through the given point.
Finding the Perpendicular Slope The slope of the given line is − 4 . The slope of a line perpendicular to this line is the negative reciprocal of − 4 , which is 4 1 .
Applying Point-Slope Form Now we use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point and m is the slope. In our case, ( x 1 , y 1 ) = ( 8 , 1 ) and m = 4 1 . Substituting these values, we get y − 1 = 4 1 ( x − 8 ) .
Converting to Slope-Intercept Form Now, we solve for y to get the equation in slope-intercept form:
y − 1 = 4 1 x − 4 1 ( 8 )
y − 1 = 4 1 x − 2
y = 4 1 x − 2 + 1
y = 4 1 x − 1
Final Answer The equation of the line is y = 4 1 x − 1 .
Examples
Imagine you're designing a ramp that needs to be perpendicular to a wall. The wall's slope is like the slope of our given line, and the point (8,1) is where the ramp needs to connect. Finding the equation of the ramp's line ensures it's perfectly perpendicular to the wall and meets the connection point correctly. This is crucial for safety and functionality in construction and design.
