Which of these points lies on the line described by the equation below?
[tex]y-4=-2(x-6)[/tex]
A. [tex](-6,-4)[/tex]
B. [tex](-4,-6)[/tex]
C. [tex](6,4)[/tex]
D. [tex](4,6)[/tex]
Answer (1)
Substitute the coordinates of each point into the equation y − 4 = − 2 ( x − 6 ) .
Check point A ( − 6 , − 4 ) : − 4 − 4 = − 2 ( − 6 − 6 ) ⇒ − 8 = 24 (False).
Check point B ( − 4 , − 6 ) : − 6 − 4 = − 2 ( − 4 − 6 ) ⇒ − 10 = 20 (False).
Check point C ( 6 , 4 ) : 4 − 4 = − 2 ( 6 − 6 ) ⇒ 0 = 0 (True).
Check point D ( 4 , 6 ) : 6 − 4 = − 2 ( 4 − 6 ) ⇒ 2 = 4 (False).
Point C ( 6 , 4 ) lies on the line, so the answer is ( 6 , 4 ) .
Explanation
Problem Analysis We are given the equation of a line: y − 4 = − 2 ( x − 6 ) , and we need to determine which of the given points lies on this line. To do this, we will substitute the coordinates of each point into the equation and see if the equation holds true.
Checking Point A Let's check point A: ( − 6 , − 4 ) . Substitute x = − 6 and y = − 4 into the equation: ( − 4 ) − 4 = − 2 (( − 6 ) − 6 ) − 8 = − 2 ( − 12 ) − 8 = 24 . This is false.
Checking Point B Now, let's check point B: ( − 4 , − 6 ) . Substitute x = − 4 and y = − 6 into the equation: ( − 6 ) − 4 = − 2 (( − 4 ) − 6 ) − 10 = − 2 ( − 10 ) − 10 = 20 . This is false.
Checking Point C Next, let's check point C: ( 6 , 4 ) . Substitute x = 6 and y = 4 into the equation: ( 4 ) − 4 = − 2 (( 6 ) − 6 ) 0 = − 2 ( 0 ) 0 = 0 . This is true.
Checking Point D Finally, let's check point D: ( 4 , 6 ) . Substitute x = 4 and y = 6 into the equation: ( 6 ) − 4 = − 2 (( 4 ) − 6 ) 2 = − 2 ( − 2 ) 2 = 4 . This is false.
Conclusion Since the equation holds true for point C ( 6 , 4 ) , this point lies on the line.
Examples
Understanding which points lie on a line is a fundamental concept in coordinate geometry and has many real-world applications. For example, if you are designing a road or a railway track, you need to ensure that the path follows a specific equation. By checking if certain points lie on the line, you can verify that the construction is accurate and meets the required specifications. This ensures safety and efficiency in transportation systems.
