What is a point-slope equation of the line with slope -13 that goes through the point $(5,7)$?
A. $y+7=-13(x+5)$
B. $y-7=-13(x-5)$
C. $y-5=-13(x-7)$
D. $y+5=-13(x+7)$
Answer (1)
Recall the point-slope form: y − y 1 = m ( x − x 1 ) .
Substitute the given slope m = − 13 and the point ( 5 , 7 ) into the formula.
Obtain the equation: y − 7 = − 13 ( x − 5 ) .
The point-slope equation of the line is y − 7 = − 13 ( x − 5 ) .
Explanation
Understanding the Problem We are given the slope of a line, m = − 13 , and a point that the line passes through, ( 5 , 7 ) . We need to find the point-slope equation of this line.
Recalling Point-Slope Form The point-slope form of a linear equation is given by: y − y 1 = m ( x − x 1 ) where m is the slope and ( x 1 , y 1 ) is a point on the line.
Substituting the Values We are given m = − 13 and ( x 1 , y 1 ) = ( 5 , 7 ) . Substituting these values into the point-slope form, we get: y − 7 = − 13 ( x − 5 )
Finding the Correct Option Comparing this equation with the given options, we see that it matches option B.
Final Answer Therefore, the point-slope equation of the line with slope -13 that goes through the point ( 5 , 7 ) is y − 7 = − 13 ( x − 5 ) .
Examples
Point-slope form is useful in many real-world scenarios. For example, if you know the rate at which a savings account is growing (slope) and the current balance (a point), you can predict the balance at any future time. Similarly, in physics, if you know the velocity of an object (slope) and its position at a certain time (a point), you can determine its position at any other time.
