Which of the following points lies on the line $y=x+1$?
(2,3)
(2,-3)
(-2,3)
(-2,-3)
Answer (2)
To determine which point lies on the line y = x + 1 , we substitute the coordinates of each point into the equation. The point ( 2 , 3 ) satisfies the equation because 3 = 2 + 1 . The other points do not satisfy the equation. Therefore, ( 2 , 3 ) is the point that lies on the line.
Explanation
Understanding the Problem We are given the equation of a line y = x + 1 and four points: ( 2 , 3 ) , ( 2 , − 3 ) , ( − 2 , 3 ) , and ( − 2 , − 3 ) . 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. We will substitute the x and y coordinates of each point into the equation and check if the equation holds true.
Checking the point (2,3) Let's check the point ( 2 , 3 ) . Substituting x = 2 and y = 3 into the equation y = x + 1 , we get 3 = 2 + 1 , which simplifies to 3 = 3 . This is true, so the point ( 2 , 3 ) lies on the line.
Checking the point (2,-3) Let's check the point ( 2 , − 3 ) . Substituting x = 2 and y = − 3 into the equation y = x + 1 , we get − 3 = 2 + 1 , which simplifies to − 3 = 3 . This is false, so the point ( 2 , − 3 ) does not lie on the line.
Checking the point (-2,3) Let's check the point ( − 2 , 3 ) . Substituting x = − 2 and y = 3 into the equation y = x + 1 , we get 3 = − 2 + 1 , which simplifies to 3 = − 1 . This is false, so the point ( − 2 , 3 ) does not lie on the line.
Checking the point (-2,-3) Let's check the point ( − 2 , − 3 ) . Substituting x = − 2 and y = − 3 into the equation y = x + 1 , we get − 3 = − 2 + 1 , which simplifies to − 3 = − 1 . This is false, so the point ( − 2 , − 3 ) does not lie on the line.
Conclusion Therefore, only the point ( 2 , 3 ) lies on the line y = x + 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 room, an architect needs to ensure that certain elements (like a support beam) align perfectly along a specified line to maintain structural integrity. By using the equation of a line and substituting the coordinates of the point, the architect can quickly verify if the element is correctly positioned. This ensures the design is both aesthetically pleasing and structurally sound, preventing potential issues down the line.
The point that lies on the line y = x + 1 is ( 2 , 3 ) . This point satisfies the equation, while the others do not. Therefore, the correct answer is ( 2 , 3 ) .
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