Find the equation of the line perpendicular to [tex]y=-\frac{1}{2} x-5[/tex] that passes through the point (2,7). Write this line in slope-intercept form.
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Answer (2)
Find the slope of the perpendicular line: m = 2 .
Use the point-slope form with the point (2, 7): y − 7 = 2 ( x − 2 ) .
Convert to slope-intercept form: y = 2 x + 3 .
The equation of the perpendicular line is y = 2 x + 3 .
Explanation
Understanding the Problem We are given the equation of a line y = − 2 1 x − 5 and a point ( 2 , 7 ) . We need to find the equation of the line that is perpendicular to the given line and passes through the given point. 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 Perpendicular Line The slope of the given line is − 2 1 . The slope of a line perpendicular to this line is the negative reciprocal of − 2 1 , which is m = − − 2 1 1 = 2 .
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 ( 2 , 7 ) and m is the slope of the perpendicular line, which is 2. Substituting the values, we get y − 7 = 2 ( x − 2 ) .
Converting to Slope-Intercept Form Next, we convert the equation to slope-intercept form, y = m x + b , by solving for y . We have y − 7 = 2 ( x − 2 ) . Simplifying, we get y − 7 = 2 x − 4 . Adding 7 to both sides, we get y = 2 x − 4 + 7 , which simplifies to y = 2 x + 3 .
Final Answer Therefore, the equation of the line perpendicular to y = − 2 1 x − 5 that passes through the point ( 2 , 7 ) is y = 2 x + 3 .
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 equation of the line representing that perpendicular garden side, ensuring it meets the fence at a perfect right angle. Understanding perpendicular lines is crucial in construction, architecture, and even creating accurate maps. This algebraic approach ensures precision in real-world applications.
The equation of the line perpendicular to y = − 2 1 x − 5 that passes through the point ( 2 , 7 ) is y = 2 x + 3 . This was derived by finding the slope of the original line, determining the negative reciprocal for the perpendicular line, using the point-slope formula, and simplifying to slope-intercept form.
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