Write the equation of a line that passes through $(-6,2)$ and is perpendicular to $y=\frac{3}{2} x+8$. Express your answer in slope-intercept form.
Answer (2)
Determine the slope of the given line: 2 3 .
Calculate the slope of the perpendicular line: − 3 2 .
Use the point-slope form with the point ( − 6 , 2 ) and the perpendicular slope: y − 2 = − 3 2 ( x + 6 ) .
Convert to slope-intercept form: y = − 3 2 x − 2 .
The equation of the line is y = − 3 2 x − 2 .
Explanation
Understanding the Problem We are given a point ( − 6 , 2 ) and a line y = 2 3 x + 8 . We need to find the equation of a line that passes through the given point and is perpendicular to the given line. The final answer must 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 Perpendicular Line The slope of the given line y = 2 3 x + 8 is 2 3 . Since we want to find a line perpendicular to this line, we need to find the negative reciprocal of the slope. The negative reciprocal of 2 3 is − 3 2 . So, the slope of the perpendicular line is − 3 2 .
Using the Point-Slope Form Now we have the slope of the perpendicular line, which is − 3 2 , and a point that the line passes through, which is ( − 6 , 2 ) . We can use the point-slope form of a line to find the equation of the line. The point-slope form 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 − 2 = − 3 2 ( x − ( − 6 )) .
Converting to Slope-Intercept Form Now we need to convert the equation to slope-intercept form, which is y = m x + b . First, we simplify the equation:
y − 2 = − 3 2 ( x + 6 )
y − 2 = − 3 2 x − 3 2 ( 6 )
y − 2 = − 3 2 x − 4
Now, we add 2 to both sides of the equation:
y = − 3 2 x − 4 + 2
y = − 3 2 x − 2
Final Answer The equation of the line that passes through ( − 6 , 2 ) and is perpendicular to y = 2 3 x + 8 is y = − 3 2 x − 2 .
Examples
Imagine you're designing a rectangular garden and one side needs to be perpendicular to an existing fence. This problem helps you determine the slope and equation of the line representing the garden's side, ensuring it's perfectly perpendicular to the fence. Understanding perpendicular lines is crucial in construction, architecture, and design, where precise angles and alignments are essential for stability and aesthetics. This algebraic approach ensures accuracy in real-world applications.
The equation of the line passing through the point (-6, 2) and perpendicular to the line y = 3/2x + 8 is y = -2/3x - 2. This is found by determining the negative reciprocal of the slope and using the point-slope form. The final result is expressed in slope-intercept form.
;
