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)
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 of 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 m = − 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 Option Comparing the simplified equation y + 3 = − 6 ( x + 9 ) with the given options, we find that it matches the fourth option. Therefore, the equation of the line is y + 3 = − 6 ( x + 9 ) .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you are tracking the depreciation of a car, you can use a linear equation to model how the car's value decreases over time. The slope would represent the rate of depreciation, and a specific point could represent the car's value at a certain age. Similarly, in physics, you can use linear equations to describe the motion of an object moving at a constant velocity. The slope would represent the velocity, and a point could represent the object's position at a specific time. These examples demonstrate how linear equations provide a powerful tool for modeling and understanding various phenomena in everyday life.
The equation of the line that passes through the point ( − 9 , − 3 ) with a slope of − 6 is given by y + 3 = − 6 ( x + 9 ) . The correct answer is option D. This can be derived from the point-slope form of a linear equation.
;
