Write an equation of the line that passes through the point $(-4,6)$ with slope -4.
A. $y+4=-4(x-6)$
B. $y-6=-4(x+4)$
C. $y+4=4(x-6)$
D. $y-6=4(x+4)$
Answer (2)
Use the point-slope form of a line: y − y 1 = m ( x − x 1 ) .
Substitute the given point ( − 4 , 6 ) and slope m = − 4 into the point-slope form: y − 6 = − 4 ( x + 4 ) .
Simplify the equation and compare it with the given options.
The equation of the line is y − 6 = − 4 ( x + 4 ) .
Explanation
Understanding the Problem We are given a point ( − 4 , 6 ) and a slope m = − 4 . We need to find the equation of the line that passes through this point with the given 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 ( − 4 , 6 ) and slope m = − 4 into the point-slope form: y − 6 = − 4 ( x − ( − 4 )) y − 6 = − 4 ( x + 4 )
Finding the Correct Option Comparing the equation y − 6 = − 4 ( x + 4 ) with the given options, we see that it matches option B.
Final Answer Therefore, the equation of the line that passes through the point ( − 4 , 6 ) with slope -4 is: y − 6 = − 4 ( x + 4 )
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you're tracking the depreciation of a car, the value of the car decreases linearly over time. If you know the initial value of the car and the rate at which it depreciates, you can use a linear equation to predict its value at any point in the future. Similarly, in physics, the relationship between distance, rate, and time for an object moving at a constant speed can be modeled using a linear equation. These equations help in making predictions and informed decisions based on available data.
The equation of the line passing through the point ( − 4 , 6 ) with a slope of -4 is y − 6 = − 4 ( x + 4 ) . This matches option B from the provided choices. Thus, the answer is option B.
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