Which point-slope equation represents a line that passes through $(3,-2)$ with a slope of $-\frac{4}{5}$?
A. $y-3=-\frac{4}{5}(x+2)$
B. $y-2=\frac{4}{5}(x-3)$
C. $y+2=-\frac{4}{5}(x-3)$
D. $y+3=\frac{1}{5}(x+2)$
Answer (2)
Recall the point-slope form: y − y 1 = m ( x − x 1 ) .
Substitute the given point ( 3 , − 2 ) and slope − 5 4 into the point-slope form: y − ( − 2 ) = − 5 4 ( x − 3 ) .
Simplify the equation: y + 2 = − 5 4 ( x − 3 ) .
The correct point-slope equation is: y + 2 = − 5 4 ( x − 3 )
Explanation
Understanding the problem We are given a point ( 3 , − 2 ) and a slope m = − 5 4 . We need to find the point-slope equation of the line.
Recalling the 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.
Substituting the values Substitute the given point ( 3 , − 2 ) and slope − 5 4 into the point-slope form: y − ( − 2 ) = − 5 4 ( x − 3 ) Simplify the equation: y + 2 = − 5 4 ( x − 3 )
Finding the correct equation Comparing the simplified equation with the given options, we find that the correct equation is: y + 2 = − 5 4 ( x − 3 )
Examples
The point-slope form is useful in various real-world scenarios, such as determining the equation of a ski slope given a point and the slope, or modeling the cost of a service that increases linearly with usage, where you know the initial cost and the rate of increase. For example, if a taxi charges a flat fee and then an additional amount per mile, the point-slope form can help you determine the total cost based on the distance traveled.
The correct point-slope equation for the line through the point ( 3 , − 2 ) with a slope of − 5 4 is given by option C: y + 2 = − 5 4 ( x − 3 ) .
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