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 (2)
The problem provides a point and a slope and asks for the equation of the line.
Use the point-slope form of a line: 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 to get the final answer: 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 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 Substitute the given point ( − 9 , − 3 ) and slope m = − 6 into the point-slope form: y − ( − 3 ) = − 6 ( x − ( − 9 ))
Simplifying the Equation Simplify the equation: y + 3 = − 6 ( x + 9 ) This matches one of the given options.
Final Answer Therefore, the equation of the line that passes through ( − 9 , − 3 ) and has a slope of -6 is: y + 3 = − 6 ( x + 9 )
Examples
Understanding linear equations is crucial in many real-world scenarios. For instance, if you are 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 rate of 6 miles per hour, the equation y + 3 = -6(x + 9) can be adapted to represent the car's position relative to a certain point, helping you predict its location at any given time. Similarly, linear equations are used in calculating simple interest, determining the cost of services based on usage, and modeling various other proportional relationships.
The equation of the line passing through ( − 9 , − 3 ) with a slope of − 6 is given by the point-slope form, which simplifies to y + 3 = − 6 ( x + 9 ) . The correct answer is option D. This equation accurately reflects the linear relationship described.
;
