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. $y-1=-\frac{1}{2}(x-4)$
C. $y-1=\frac{1}{2}(x-4)$
D. $y-1=2(x-4)$
Answer (1)
The problem requires finding the equation of a line parallel to a given line and passing through the point (4, 1).
The point-slope form of a line is y − y 1 = m ( x − x 1 ) .
Parallel lines have the same slope.
Assuming the original line has a slope of 2, the equation of the parallel 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 y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line. Parallel lines have the same slope.
Finding the Slope First, we need to determine the slope of the given line. The options are in point-slope form, so we can identify the slope from each option. The options are:
y − 1 = − 2 ( x − 4 ) : Slope m = − 2
y − 1 = − 2 1 ( x − 4 ) : Slope m = − 2 1
y − 1 = 2 1 ( x − 4 ) : Slope m = 2 1
y − 1 = 2 ( x − 4 ) : Slope m = 2
Since we are looking for the equation of a line that passes through the point ( 4 , 1 ) , we can plug this point into the point-slope form y − y 1 = m ( x − x 1 ) , which gives us y − 1 = m ( x − 4 ) . We need to find which of the given options has the correct slope for a line parallel to the given line.
Finding the Parallel Line Equation The question does not provide the 'given line'. However, the options provide possible equations for the line we are looking for. Since the line passes through ( 4 , 1 ) , the equation will be of the form y − 1 = m ( x − 4 ) . We need to determine which of the given options is the correct one.
Without knowing the original line, we cannot determine which slope is the correct one for a parallel line. However, if we assume that one of the options is the original line, then the parallel line will have the same equation. Therefore, we look for the option that has the form y − 1 = m ( x − 4 ) . All options have this form. We need to know the original line to determine the slope of the parallel line.
Let's assume the original line is y = 2 x + 5 . A parallel line will have the same slope, m = 2 . The equation of the parallel line passing through ( 4 , 1 ) is y − 1 = 2 ( x − 4 ) . This matches option 4.
Let's assume the original line is y = − 2 x + 5 . A parallel line will have the same slope, m = − 2 . The equation of the parallel line passing through ( 4 , 1 ) is y − 1 = − 2 ( x − 4 ) . This matches option 1.
Let's assume the original line is y = 2 1 x + 5 . A parallel line will have the same slope, m = 2 1 . The equation of the parallel line passing through ( 4 , 1 ) is y − 1 = 2 1 ( x − 4 ) . This matches option 3.
Let's assume the original line is y = − 2 1 x + 5 . A parallel line will have the same slope, m = − 2 1 . The equation of the parallel line passing through ( 4 , 1 ) is y − 1 = − 2 1 ( x − 4 ) . This matches option 2.
Conclusion Since the problem statement is ambiguous and does not provide the original line, we cannot determine the correct answer. However, if we assume that the question is asking which of the given options represents a line passing through ( 4 , 1 ) , then all options are valid. If we assume that one of the options is the original line, then the parallel line will have the same equation. Without more information, we cannot determine the correct answer. However, if the original line is y = 2 x + 5 , then the parallel line is y − 1 = 2 ( x − 4 ) .
Final Answer Assuming the original line has a slope of 2, the equation of the parallel line in point-slope form is y − 1 = 2 ( x − 4 ) .
Examples
Understanding parallel lines is crucial in architecture and design. For instance, when designing a building, ensuring that walls are parallel is essential for structural integrity and aesthetic appeal. If a wall needs to be parallel to another and pass through a specific point, the principles used in this problem can be applied to determine the equation representing the new wall's alignment. This ensures that the design is both functional and visually harmonious.
