A line intersects the points $(8,2)$ and $(12,-10)$.
$m=-3$
Write the equation of this line in point-slope form using the point $(8,2)$.
y-[?]=\square(x-\square)
Answer (2)
The equation of the line in point-slope form using the point ( 8 , 2 ) is found by substituting the given point and slope into the point-slope form equation. The equation is y − 2 = − 3 ( x − 8 ) .
Explanation
Understanding the Problem We are given a line that intersects the points ( 8 , 2 ) and ( 12 , − 10 ) , and we are also given that the slope of the line is m = − 3 . We want to write the equation of this line in point-slope form using the point ( 8 , 2 ) . The point-slope form of a line 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 Values We are given the point ( 8 , 2 ) , so x 1 = 8 and y 1 = 2 . We are also given the slope m = − 3 . Substituting these values into the point-slope form, we get:
y − 2 = − 3 ( x − 8 )
Writing the Equation Now we fill in the blanks in the given equation y − [ ?] = □ ( x − □ ) with the corresponding values from the point-slope form. We have:
y − 2 = − 3 ( x − 8 )
So, the equation is y − 2 = − 3 ( x − 8 ) .
Examples
Point-slope form is useful in many real-world scenarios. For example, if you know the rate at which you are saving money (slope) and how much you have saved at a certain time (point), you can use the point-slope form to determine how much money you will have saved at any given time in the future. This is a practical application of linear equations in personal finance.
The equation of the line in point-slope form using the point ( 8 , 2 ) is y − 2 = − 3 ( x − 8 ) . This is based on substituting the given point and slope into the point-slope formula. The identified values are y 1 = 2 and x 1 = 8 .
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