What is the equation, in point-slope form, of the line that is parallel to the given line and passes through the point $(4,1)$?
A. $y-1=-2(x-4)$
B. $y-1=-\frac{1}{2}(x-1)$
C. $y-1=\frac{1}{2}(x-4)$
D. $y-1=2(x-4)$
Answer (1)
The point-slope form of a line is y − y 1 = m ( x − x 1 ) .
Identify the point ( x 1 , y 1 ) = ( 4 , 1 ) .
Assume the question intended to ask which line has a slope of 2.
The equation of the line is y − 1 = 2 ( x − 4 ) .
Explanation
Understanding the Problem We are given a point ( 4 , 1 ) and need to find the equation of a line in point-slope form that is parallel to a given line. 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. Parallel lines have the same slope. We need to identify the line that has the same slope as the line passing through the point ( 4 , 1 ) .
Analyzing the Options The given options are:
a) y − 1 = − 2 ( x − 4 ) b) y − 1 = − 2 1 ( x − 1 ) c) y − 1 = 2 1 ( x − 4 ) d) y − 1 = 2 ( x − 4 )
We are looking for the equation of a line that passes through the point ( 4 , 1 ) . All the options are in point-slope form with the point ( 4 , 1 ) or ( 1 , 1 ) . We need to find the line that is parallel to the given line.
Identifying the Correct Slope The point-slope form is y − y 1 = m ( x − x 1 ) . Since the line passes through ( 4 , 1 ) , the equation will be of the form y − 1 = m ( x − 4 ) .
Now, we need to determine which of the given options represents a line parallel to the line passing through ( 4 , 1 ) . This means we need to find the correct slope m .
Looking at the options:
a) y − 1 = − 2 ( x − 4 ) has slope m = − 2 and passes through ( 4 , 1 ) .
b) y − 1 = − 2 1 ( x − 1 ) has slope m = − 2 1 and passes through ( 1 , 1 ) .
c) y − 1 = 2 1 ( x − 4 ) has slope m = 2 1 and passes through ( 4 , 1 ) .
d) y − 1 = 2 ( x − 4 ) has slope m = 2 and passes through ( 4 , 1 ) .
Finding the Correct Equation We are looking for the equation of the line that is parallel to the given line. However, there is no given line in the problem description. Let's assume that the question meant to ask which of the options is a line that passes through the point (4,1). In that case, options a, c, and d are all possible answers. However, option b passes through (1,1), so it is not the correct answer.
If we assume that the question meant to ask which of the options is a line that passes through the point (4,1) and has a slope of 2, then the answer would be option d.
Final Answer Without a specific "given line" to compare to, we can only determine which of the provided options correctly represents a line passing through the point (4,1) in point-slope form. Options a, c, and d all satisfy this condition. However, if we assume the question intended to ask which line has a slope of 2 and passes through (4,1), then option d is the correct answer.
Therefore, assuming the intended question was: "What is the equation, in point-slope form, of the line that passes through the point ( 4 , 1 ) and has a slope of 2?"
The answer is: y − 1 = 2 ( x − 4 )
Examples
Understanding point-slope form helps in various real-world scenarios. For instance, if you know the rate at which a savings account grows (slope) and the initial amount (a point), you can predict the balance at any time. Similarly, in physics, knowing the velocity (slope) of an object and its position at a specific time (a point) allows you to determine its location at any other time. This concept is fundamental in modeling linear relationships and making predictions based on available data.
