Which of the following points lies on the line [tex]$3 x-y=-1$[/tex]?
A. [tex]$(-1,3)$[/tex]
B. [tex]$(1,2)$[/tex]
C. [tex]$(1,4)$[/tex]
D. [tex]$(2,5)$[/tex]
Answer (1)
Substitute each point's coordinates into the equation 3 x − y = − 1 .
Check if the equation holds true for each point.
The point ( − 1 , 3 ) gives 3 ( − 1 ) − 3 = − 6 = − 1 .
The point ( 1 , 4 ) gives 3 ( 1 ) − 4 = − 1 , which satisfies the equation, so the final answer is ( 1 , 4 ) .
Explanation
Understanding the Problem We are given the equation of a line 3 x − y = − 1 and four points: ( − 1 , 3 ) , ( 1 , 2 ) , ( 1 , 4 ) , and ( 2 , 5 ) . We need to determine which of these points lies on the given line. A point lies on a line if its coordinates satisfy the equation of the line.
Checking the Points Let's check each point to see if it satisfies the equation 3 x − y = − 1 .
Checking (-1, 3)
For the point ( − 1 , 3 ) , we substitute x = − 1 and y = 3 into the equation: 3 ( − 1 ) − 3 = − 3 − 3 = − 6 Since − 6 e q − 1 , the point ( − 1 , 3 ) does not lie on the line.
Checking (1, 2)
For the point ( 1 , 2 ) , we substitute x = 1 and y = 2 into the equation: 3 ( 1 ) − 2 = 3 − 2 = 1 Since 1 e q − 1 , the point ( 1 , 2 ) does not lie on the line.
Checking (1, 4)
For the point ( 1 , 4 ) , we substitute x = 1 and y = 4 into the equation: 3 ( 1 ) − 4 = 3 − 4 = − 1 Since − 1 = − 1 , the point ( 1 , 4 ) lies on the line.
Checking (2, 5)
For the point ( 2 , 5 ) , we substitute x = 2 and y = 5 into the equation: 3 ( 2 ) − 5 = 6 − 5 = 1 Since 1 e q − 1 , the point ( 2 , 5 ) does not lie on the line.
Conclusion Therefore, the point ( 1 , 4 ) is the only point among the given options that lies on the line 3 x − y = − 1 .
Examples
In architecture, determining if a point lies on a line is crucial for designing structures. For example, when planning the layout of a building, architects need to ensure that certain points (like corners of rooms or support columns) align perfectly with predefined lines (like walls or structural beams). By using the equation of a line, they can verify whether specific coordinates meet the required alignment, ensuring the structural integrity and aesthetic appeal of the design. This principle extends to various fields, including urban planning, civil engineering, and even computer graphics, where precise alignment is essential for creating accurate models and designs.
