Find the equation of the line that is perpendicular to [tex]y=\frac{1}{6} x+3[/tex] and contains the point $(-3,23)$.
[tex]y=[?] x+[] [/tex]
Answer (2)
Find the slope of the given line: m 1 = 6 1 .
Calculate the slope of the perpendicular line: m 2 = − m 1 1 = − 6 .
Use the point-slope form with the point ( − 3 , 23 ) : y − 23 = − 6 ( x + 3 ) .
Convert to slope-intercept form: y = − 6 x + 5 , so the final answer is y = − 6 x + 5 .
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. The given line is y = 6 1 x + 3 , and the point is ( − 3 , 23 ) .
Finding the Slope of the Given Line First, we need to find the slope of the given line. The equation is in slope-intercept form, y = m x + b , where m is the slope. In this case, the slope of the given line is 6 1 .
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. So, the slope of the perpendicular line is − 6 1 1 = − 6 .
Using the Point-Slope Form Now we 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 given point. We have m = − 6 and ( x 1 , y 1 ) = ( − 3 , 23 ) . Plugging these values into the point-slope form, we get y − 23 = − 6 ( x − ( − 3 )) , which simplifies to y − 23 = − 6 ( x + 3 ) .
Converting to Slope-Intercept Form Now, we convert the equation to slope-intercept form, y = m x + b . Starting with y − 23 = − 6 ( x + 3 ) , we distribute the − 6 to get y − 23 = − 6 x − 18 . Then, we add 23 to both sides to isolate y : y = − 6 x − 18 + 23 , which simplifies to y = − 6 x + 5 .
Final Answer The equation of the line perpendicular to y = 6 1 x + 3 and passing through the point ( − 3 , 23 ) is y = − 6 x + 5 .
Examples
Understanding perpendicular lines is crucial in many real-world applications, such as architecture and construction. For example, when designing a building, ensuring that walls are perpendicular to the ground is essential for stability. Similarly, in navigation, understanding perpendicular paths helps in determining the shortest distance between two points or avoiding obstacles. The principles of perpendicular lines also apply in computer graphics, where they are used to create realistic images and animations.
The perpendicular line to y = 6 1 x + 3 through the point ( − 3 , 23 ) has the equation y = − 6 x + 5 . First, we determined the slope of the given line and found the negative reciprocal for the perpendicular slope. Using point-slope form, we derived the final equation.
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