Which point lies on the line described by the equation below? [tex]y+8=4(x-5)[/tex]
A. [tex](5,8)[/tex]
B. [tex](5,0)[/tex]
C. [tex](-5,-8)[/tex]
D. [tex](5,-8)[/tex]
E. [tex](4,-8)[/tex]
F. [tex](4,5)[/tex]

Substitute the coordinates of each point into the equation y + 8 = 4 ( x − 5 ) .
Check if the equation holds true for each point.
Point A ( 5 , 8 ) : 16 = 0"> 8 + 8 = 4 ( 5 − 5 ) ′ => 16 = 0 (False).
Point D ( 5 , − 8 ) : 0 = 0"> − 8 + 8 = 4 ( 5 − 5 ) ′ => 0 = 0 (True). Therefore, the answer is ( 5 , − 8 ) ​ .

Explanation

Problem Analysis We are given the equation of a line y + 8 = 4 ( x − 5 ) and six points. Our goal is to find which of these points lies on the line. A point lies on the line 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.

Checking Point A Let's check point A ( 5 , 8 ) . Substituting x = 5 and y = 8 into the equation, we get: 8 + 8 = 4 ( 5 − 5 ) 16 = 4 ( 0 ) 16 = 0 This is false, so point A does not lie on the line.

Checking Point B Let's check point B ( 5 , 0 ) . Substituting x = 5 and y = 0 into the equation, we get: 0 + 8 = 4 ( 5 − 5 ) 8 = 4 ( 0 ) 8 = 0 This is false, so point B does not lie on the line.

Checking Point C Let's check point C ( − 5 , − 8 ) . Substituting x = − 5 and y = − 8 into the equation, we get: − 8 + 8 = 4 ( − 5 − 5 ) 0 = 4 ( − 10 ) 0 = − 40 This is false, so point C does not lie on the line.

Checking Point D Let's check point D ( 5 , − 8 ) . Substituting x = 5 and y = − 8 into the equation, we get: − 8 + 8 = 4 ( 5 − 5 ) 0 = 4 ( 0 ) 0 = 0 This is true, so point D lies on the line.

Final Answer Since we found a point that lies on the line, we can stop here. Point D ( 5 , − 8 ) satisfies the equation y + 8 = 4 ( x − 5 ) . Therefore, point D lies on the line.


Examples
Understanding which points lie on a line is crucial in various real-world applications. For instance, in navigation, determining if a ship's coordinates lie on a planned route ensures it stays on course. Similarly, in computer graphics, verifying if a pixel lies on a line segment is essential for rendering images accurately. In engineering, this concept helps in designing structures where certain points must align along a specific line for stability and functionality.

Answered by GinnyAnswer | 2025-07-04

After substituting the coordinates of the points into the equation y + 8 = 4 ( x − 5 ) , only Point D (5, -8) satisfies the equation. Therefore, Point D lies on the line. The answer is (5, -8).
;

Answered by Anonymous | 2025-07-10