What is the equation of the line in slope-intercept form that is perpendicular to the line [tex]y=\frac{3}{4} x-2[/tex] and passes through the point (-12,10)?
A. [tex]y=-\frac{4}{3} x-6[/tex]
B. [tex]y=-\frac{4}{3} x+6[/tex]
C. [tex]y=\frac{4}{3} x+26[/tex]
D. [tex]y=\frac{4}{3} x+10[/tex]
Answer (2)
Find the slope of the given line: The slope of y = 4 3 x − 2 is 4 3 .
Find the slope of the perpendicular line: The slope is − 3 4 .
Use the point-slope form: y − 10 = − 3 4 ( x + 12 ) .
Convert to slope-intercept form: y = − 3 4 x − 6 .
The equation of the line is y = − 3 4 x − 6 .
Explanation
Understanding the Problem The problem asks us to find the equation of a line that is perpendicular to a given line and passes through a specific point. We need to find the slope of the perpendicular line and then use the point-slope form to find the equation of the line. Finally, we will convert the equation to slope-intercept form.
Finding the Slope of the Given Line The given line is y = 4 3 x − 2 . The slope of this line is 4 3 .
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. Therefore, the slope of the perpendicular line is − 3 4 .
Using the Point-Slope Form We are given the point ( − 12 , 10 ) that the perpendicular line passes through. We can use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is the point. Substituting the values, we get y − 10 = − 3 4 ( x − ( − 12 )) .
Converting to Slope-Intercept Form Now, we simplify the equation and convert it to slope-intercept form ( y = m x + b ).
y − 10 = − 3 4 ( x + 12 )
y − 10 = − 3 4 x − 3 4 ( 12 )
y − 10 = − 3 4 x − 16
y = − 3 4 x − 16 + 10
y = − 3 4 x − 6
Final Answer The equation of the line in slope-intercept form that is perpendicular to the line y = 4 3 x − 2 and passes through the point ( − 12 , 10 ) is y = − 3 4 x − 6 .
Examples
Understanding perpendicular lines is crucial in various real-world applications, such as architecture and navigation. For example, when designing a building, architects need to ensure that walls are perpendicular to the ground for stability. Similarly, in navigation, understanding perpendicular paths helps in determining the shortest distance between two points or avoiding obstacles.
The equation of the line perpendicular to y = 4 3 x − 2 passing through the point ( − 12 , 10 ) is y = − 3 4 x − 6 . This corresponds to option A.
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