Write an equation for a line perpendicular to [tex]$y=-5 x+2$[/tex] and passing through the point (-15,1).
[tex]$y=$[/tex]
Answer (1)
Find the slope of the given line: m = − 5 .
Calculate the slope of the perpendicular line: m ⊥ = 5 1 = 0.2 .
Use the point-slope form with the point ( − 15 , 1 ) : y − 1 = 0.2 ( x + 15 ) .
Convert to slope-intercept form: y = 0.2 x + 4 . The final answer is y = 0.2 x + 4 .
Explanation
Understanding the Problem We are given the equation of a line y = − 5 x + 2 and a point ( − 15 , 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 Slope of the Given Line The given line is in slope-intercept form, y = m x + b , where m is the slope. In this case, the slope of the given line is m = − 5 .
Finding the Slope of the Perpendicular Line The slope of a line perpendicular to the given line is the negative reciprocal of the slope of the given line. If the slope of the given line is m , then the slope of the perpendicular line is − m 1 . Therefore, the slope of the perpendicular line is − − 5 1 = 5 1 = 0.2 .
Using the Point-Slope Form Now we use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point and m is the slope of the perpendicular line. We are given the point ( − 15 , 1 ) and the slope of the perpendicular line is 0.2 . Substituting these values into the point-slope form, we get:
y − 1 = 0.2 ( x − ( − 15 )) y − 1 = 0.2 ( x + 15 )
Converting to Slope-Intercept Form Now, we convert the equation to slope-intercept form, y = m x + b , to get the final equation:
y − 1 = 0.2 x + 0.2 ( 15 ) y − 1 = 0.2 x + 3 y = 0.2 x + 3 + 1 y = 0.2 x + 4
Final Answer Therefore, the equation of the line perpendicular to y = − 5 x + 2 and passing through the point ( − 15 , 1 ) is y = 0.2 x + 4 .
Examples
Understanding perpendicular lines is crucial in various real-world applications. For instance, architects and engineers use this concept to ensure that walls are built at perfect right angles, providing structural stability to buildings. Similarly, in navigation, understanding perpendicular paths helps ships and airplanes maintain safe distances and avoid collisions. In computer graphics, perpendicularity is used to create realistic lighting and shadows, enhancing the visual experience.
