A line intersects the points $(6,-4)$ and $(7,4)$.
Write the equation of this line in point-slope form using the point $(6,-4)$.
$y-[?]=\square(x-\square)$
Answer (1)
Identify the point-slope form of a line: y − y 1 = m ( x − x 1 ) .
Substitute the given point ( 6 , − 4 ) and slope m = 8 into the point-slope form: y − ( − 4 ) = 8 ( x − 6 ) .
Simplify the equation: y + 4 = 8 ( x − 6 ) .
The equation of the line in point-slope form is y + 4 = 8 ( x − 6 ) .
Explanation
Understanding the Problem We are given that a line passes through the point ( 6 , − 4 ) and has a slope of m = 8 . We want to write the equation of this line in point-slope form.
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 the Values We are given the point ( 6 , − 4 ) , so x 1 = 6 and y 1 = − 4 . We are also given the slope m = 8 . Substituting these values into the point-slope form, we get: y − ( − 4 ) = 8 ( x − 6 ) Simplifying, we have: y + 4 = 8 ( x − 6 )
Final Answer Thus, the equation of the line in point-slope form is y + 4 = 8 ( x − 6 ) .
Examples
Point-slope form is useful in many real-world applications. For example, if you know the rate at which a savings account is growing (the slope) and the amount in the account at a particular time (a point), you can use the point-slope form to determine the amount in the account at any other time. Let's say you start with $100 in your savings account, and it grows at a rate of 10 p erye a r . T h e p o in t − s l o p e f or m c anh e lp yo u p re d i c t h o w m u c hm o n eyyo u ′ ll ha v e in t h e f u t u re . I n t hi sc a se , t h ee q u a t i o n w o u l d b e y - 100 = 10(x - 0) , w h ere y i s t h e am o u n t in t h e a cco u n t , an d x$ is the number of years from the start.
