What is the equation, in slope-intercept form, of the line that is perpendicular to the given line and passes through the point $(2,-1)$?
$y=-\frac{1}{3} x-\frac{1}{3}$
$y=-\frac{1}{3} x-\frac{5}{3}$
$y=3 x-3$
$y=3 x-7$
Answer (2)
Identify the slope of the given line: m 1 = − 3 1 .
Calculate the slope of the perpendicular line: m 2 = − m 1 1 = 3 .
Use the point-slope form with the point ( 2 , − 1 ) and slope 3 : y − ( − 1 ) = 3 ( x − 2 ) .
Convert to slope-intercept form: y = 3 x − 7 , so the final answer is y = 3 x − 7 .
Explanation
Understanding the Problem We are given a line and a point, and we want 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 y = − 3 1 x − 3 1 . The slope of this line is − 3 1 .
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 m = − − 3 1 1 = 3 .
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 ( x 1 , y 1 ) is the given point and m is the slope. We are given the point ( 2 , − 1 ) , so x 1 = 2 and y 1 = − 1 . We found that the slope of the perpendicular line is m = 3 . Substituting these values into the point-slope form, we get y − ( − 1 ) = 3 ( x − 2 ) .
Converting to Slope-Intercept Form Now we convert the equation to slope-intercept form, which is y = m x + b . We have y − ( − 1 ) = 3 ( x − 2 ) , which simplifies to y + 1 = 3 x − 6 . Subtracting 1 from both sides, we get y = 3 x − 7 .
Final Answer The equation of the line that is perpendicular to the given line and passes through the point ( 2 , − 1 ) is y = 3 x − 7 .
Examples
Understanding perpendicular lines is crucial in architecture and construction. For example, when designing a building, ensuring walls are perpendicular to the ground is essential for stability. The equation of a line perpendicular to another can help architects calculate the precise angles and slopes needed for structural integrity. Similarly, in urban planning, knowing how streets intersect at right angles helps in designing efficient and safe road layouts. This concept ensures buildings are square and roads are safely aligned.
The equation of the line that is perpendicular to the given line and passes through the point (2, -1) is y = 3 x − 7 . This is derived by finding the negative reciprocal of the slope of the given line and using the point-slope form. Therefore, the correct answer is y = 3 x − 7 .
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