Which of the following points lies on the line [tex]$3 x-y=-1$[/tex]?

(-1,3)
(1,2)
(1,4)
(2,5)

To determine which point lies on the line 3 x − y = − 1 , we substitute the coordinates of each point into the equation.

For ( − 1 , 3 ) : 3 ( − 1 ) − 3 = − 6 e q − 1 .
For ( 1 , 2 ) : 3 ( 1 ) − 2 = 1 e q − 1 .
For ( 1 , 4 ) : 3 ( 1 ) − 4 = − 1 , which satisfies the equation.
For ( 2 , 5 ) : 3 ( 2 ) − 5 = 1 e q − 1 .
Therefore, the point ( 1 , 4 ) lies on the line. ( 1 , 4 ) ​

Explanation

Problem Analysis We are given the equation of a line 3 x − y = − 1 and four points: ( − 1 , 3 ) , ( 1 , 2 ) , ( 1 , 4 ) , and ( 2 , 5 ) . Our goal is to determine which of these points lies on the given line. A point lies on a line if and only if its coordinates satisfy the equation of the line. We will substitute the coordinates of each point into the equation and check if the equation holds true.

Testing Point (-1, 3) Let's test the first point ( − 1 , 3 ) . Substituting x = − 1 and y = 3 into the equation 3 x − y = − 1 , we get: 3 ( − 1 ) − 3 = − 3 − 3 = − 6 Since − 6 e q − 1 , the point ( − 1 , 3 ) does not lie on the line.

Testing Point (1, 2) Now let's test the second point ( 1 , 2 ) . Substituting x = 1 and y = 2 into the equation 3 x − y = − 1 , we get: 3 ( 1 ) − 2 = 3 − 2 = 1 Since 1 e q − 1 , the point ( 1 , 2 ) does not lie on the line.

Testing Point (1, 4) Next, let's test the third point ( 1 , 4 ) . Substituting x = 1 and y = 4 into the equation 3 x − y = − 1 , we get: 3 ( 1 ) − 4 = 3 − 4 = − 1 Since − 1 = − 1 , the point ( 1 , 4 ) lies on the line.

Testing Point (2, 5) Finally, let's test the fourth point ( 2 , 5 ) . Substituting x = 2 and y = 5 into the equation 3 x − y = − 1 , we get: 3 ( 2 ) − 5 = 6 − 5 = 1 Since 1 e q − 1 , the point ( 2 , 5 ) does not lie on the line.

Conclusion Therefore, the only point that lies on the line 3 x − y = − 1 is ( 1 , 4 ) .


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 precisely with predefined lines (like walls or structural beams). By using the equation of a line and substituting the coordinates of the points, they can verify if the points lie on the intended lines, ensuring the structural integrity and aesthetic appeal of the building. This ensures that all elements are correctly positioned according to the design specifications.

Answered by GinnyAnswer | 2025-07-08