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)
To determine which point lies on the line 3 x − y = − 1 , we substitute the coordinates of each point into the equation. The point ( 1 , 4 ) satisfies the equation since 3 ( 1 ) − 4 = − 1 . Therefore, the point ( 1 , 4 ) lies on the line. ( 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 the 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)
Point ( − 1 , 3 ) : 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)
Point ( 1 , 2 ) : 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)
Point ( 1 , 4 ) : 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)
Point ( 2 , 5 ) : 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 urban planning, determining if a location lies on a specific transportation route (like a bus line or a subway line) is a common problem. This is similar to checking if a point lies on a line. For example, if a bus route is defined by the equation 3 x − y = − 1 , then a new building at coordinates ( 1 , 4 ) would be directly on the bus route, making it easily accessible for residents. This helps planners make informed decisions about infrastructure and development.
