Write an equation of the line that passes through the point $(5,-8)$ with slope 5.
A. $y-5=5(x+8)$
B. $y+8=5(x-5)$
C. $y-5=-5(x+8)$
D. $y+8=-5(x-5)$
Answer (2)
Use the point-slope form of a line: y − y 1 = m ( x − x 1 ) .
Substitute the given point ( 5 , − 8 ) and slope 5 into the point-slope form: y − ( − 8 ) = 5 ( x − 5 ) .
Simplify the equation: y + 8 = 5 ( x − 5 ) .
The equation of the line is y + 8 = 5 ( x − 5 ) .
Explanation
Understanding the Problem We are given a point ( 5 , − 8 ) and a slope m = 5 . We need to find the equation of the line that passes through this point with the given slope.
Using Point-Slope Form The point-slope form of a linear equation is given by: y − y 1 = m ( x − x 1 ) where ( x 1 , y 1 ) is a point on the line and m is the slope of the line.
Substituting Values Substitute the given point ( 5 , − 8 ) and slope 5 into the point-slope form: y − ( − 8 ) = 5 ( x − 5 ) y + 8 = 5 ( x − 5 )
Comparing with Options Now, we compare the equation y + 8 = 5 ( x − 5 ) with the given options: A. y − 5 = 5 ( x + 8 ) B. y + 8 = 5 ( x − 5 ) C. y − 5 = − 5 ( x + 8 ) D. y + 8 = − 5 ( x − 5 ) Option B matches the equation we derived.
Final Answer Therefore, the equation of the line that passes through the point ( 5 , − 8 ) with slope 5 is y + 8 = 5 ( x − 5 ) .
Examples
Imagine you're designing a ramp for a skateboard park. You know the ramp needs to pass through a certain point and have a specific slope for safety and usability. Using the point-slope form, you can determine the exact equation of the ramp's surface, ensuring it meets the required specifications. This ensures a smooth and predictable ride for skateboarders. The point-slope form is a fundamental tool in engineering and design for creating linear structures that meet specific criteria.
The equation of the line that passes through the point ( 5 , − 8 ) with a slope of 5 is given by the equation y + 8 = 5 ( x − 5 ) . This corresponds to Option B of the multiple-choice answers provided. Thus, Option B is the correct choice.
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