Which equation represents a line that passes through $(-9,-3)$ and has a slope of $-6$?
A. $y-9=-6(x-3)$
B. $y+9=-6(x+3)$
C. $y-3=-6(x-9)$
D. $y+3=-6(x+9)$
Answer (1)
Use the point-slope form of a linear equation: y − y 1 = m ( x − x 1 ) .
Substitute the given point ( − 9 , − 3 ) and slope − 6 into the point-slope form: y − ( − 3 ) = − 6 ( x − ( − 9 )) .
Simplify the equation: y + 3 = − 6 ( x + 9 ) .
The equation that represents the line is y + 3 = − 6 ( x + 9 ) .
Explanation
Understanding the Problem We are given a point ( − 9 , − 3 ) and a slope m = − 6 . We need to find the equation of the line that passes through this point and has this slope.
Using 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 of the line.
Substituting Values Substitute the given point ( − 9 , − 3 ) and slope − 6 into the point-slope form: y − ( − 3 ) = − 6 ( x − ( − 9 ))
Simplifying the Equation Simplify the equation: y + 3 = − 6 ( x + 9 )
Finding the Correct Equation Comparing the simplified equation with the given options, we find that the correct equation is: y + 3 = − 6 ( x + 9 )
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you're tracking the distance a car travels over time at a constant speed, you can use a linear equation to model this relationship. If a car starts 3 miles from your home and travels away at a constant speed of 6 miles per hour, the equation y + 3 = -6(x + 9) can be adapted to represent the car's position relative to a reference point. In this case, it helps to understand how changes in time (x) affect the car's distance (y) from your home, allowing you to predict its location at any given time.
