What is the slope-intercept form of the equation of the line that passes through the point $(-6,1)$ and is perpendicular to the graph of $2x+3y=-5$?
A. $y=-\frac{3}{2}x-8$
B. $y=-\frac{3}{2}x+1$
C. $y=\frac{3}{2}x+1$
D. $y=\frac{3}{2}x+10
Answer (2)
Find the slope of the given line by converting it to slope-intercept form: y = − 3 2 x − 3 5 , so m 1 = − 3 2 .
Determine the slope of the perpendicular line: m 2 = − m 1 1 = 2 3 .
Use the point-slope form with the point ( − 6 , 1 ) and the perpendicular slope: y − 1 = 2 3 ( x + 6 ) .
Convert to slope-intercept form: y = 2 3 x + 10 , so the final answer is y = 2 3 x + 10 .
Explanation
Understanding the Problem We are given a point ( − 6 , 1 ) and a line 2 x + 3 y = − 5 . We need to find the equation of the line that passes through the given point and is perpendicular to the given line. The equation should be in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Finding the Slope of the Given Line First, we need to find the slope of the given line. To do this, we can rewrite the equation in slope-intercept form:
2 x + 3 y = − 5
3 y = − 2 x − 5
y = − 3 2 x − 3 5
So, the slope of the given line is m 1 = − 3 2 .
Finding the Slope of the Perpendicular Line Next, we need to find the slope of the line perpendicular to the given line. The slope of a perpendicular line is the negative reciprocal of the original line's slope. Therefore, the slope of the perpendicular line is:
m 2 = − m 1 1 = − − 3 2 1 = 2 3
Using the Point-Slope Form Now, we can 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 ( − 6 , 1 ) and m is the slope of the perpendicular line, which is 2 3 .
y − 1 = 2 3 ( x − ( − 6 ))
y − 1 = 2 3 ( x + 6 )
Converting to Slope-Intercept Form Finally, we need to convert the equation from point-slope form to slope-intercept form, y = m x + b , by solving for y :
y − 1 = 2 3 ( x + 6 )
y − 1 = 2 3 x + 2 3 ⋅ 6
y − 1 = 2 3 x + 9
y = 2 3 x + 9 + 1
y = 2 3 x + 10
Final Answer The slope-intercept form of the equation of the line that passes through the point ( − 6 , 1 ) and is perpendicular to the graph of 2 x + 3 y = − 5 is y = 2 3 x + 10 .
Examples
Understanding perpendicular lines is crucial in various real-world applications. For instance, architects use this concept to design buildings, ensuring walls are perfectly vertical to the ground for stability. Similarly, in navigation, understanding perpendicular paths helps ships and airplanes maintain safe routes, avoiding collisions. In sports, like tennis or basketball, players use angles and perpendicular movements to optimize their shots and strategies. These examples demonstrate how a solid grasp of perpendicularity enhances precision and safety in diverse fields.
The slope-intercept form of the line that is perpendicular to 2 x + 3 y = − 5 and passes through the point ( − 6 , 1 ) is y = 2 3 x + 10 . Therefore, the correct option is D.
;
