Write an equation in point-slope form for the line through the given point with the given slope.
$(8,-3) ; m=-\frac{1}{4}$
A. $y-3=-\frac{1}{4}(x+8)$
B. $y-3=-\frac{1}{4}(x-8)$
C. $y+3=-\frac{1}{4}(x-8)$
Answer (1)
The point-slope form of a line is y − y 1 = m ( x − x 1 ) .
Substitute the given point ( 8 , − 3 ) and slope m = − 4 1 into the point-slope form.
This gives y − ( − 3 ) = − 4 1 ( x − 8 ) .
Simplify to get the final equation: y + 3 = − 4 1 ( x − 8 ) .
Explanation
Understanding the Problem We are given a point ( 8 , − 3 ) and a slope m = − 4 1 . We need to write the equation of the line in point-slope form.
Recalling Point-Slope Form The point-slope form of a line is given by the equation: 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 We are given the point ( 8 , − 3 ) , so x 1 = 8 and y 1 = − 3 . We are also given the slope m = − 4 1 . Substituting these values into the point-slope form, we get: y − ( − 3 ) = − 4 1 ( x − 8 ) Simplifying the equation, we have: y + 3 = − 4 1 ( x − 8 )
Final Equation Therefore, the equation of the line in point-slope form is: y + 3 = − 4 1 ( x − 8 )
Examples
Point-slope form is useful in various real-world scenarios. For example, if you know the rate at which a savings account is growing (the slope) and the amount in the account at a specific time (a point), you can use the point-slope form to determine the amount in the account at any other time. Similarly, in physics, if you know the velocity of an object at a certain time and the constant rate of acceleration, you can determine the object's velocity at any other time using the point-slope form.
