Question

2x+3y=7

10x+15y=35
For each real number r, which of the following points lies on the graph of each equation in the xy -plane for the given system?
A) ( r/5 +7,- r/5 +35)
B) (- 3r/2 + 7/2 ,r)
C) (r, 2r/3 + 7/3 )
D) (r,- 3r/2 + 7/2 )

Answer 1

To determine which point lies on the graph of each equation in the given system, substitute the coordinates of each point into the equations and check if both equations are true. Options A, B, and D are valid points for both equations.

Step-by-step explanation:

To determine which point lies on the graph of each equation in the given system, we need to substitute the coordinates of each point into the two equations and check if both equations are true for that point. Let's evaluate each option:

1. Option A: Substituting the coordinates (r/5 +7, -r/5 +35) into the first equation, we get 2(r/5 +7) +3(-r/5 +35) = 7, which simplifies to r + 5 = 7. This equation is true, so the point lies on the graph of the first equation. Substituting the same coordinates into the second equation, we get 10(r/5 +7) +15(-r/5 +35) = 35, which simplifies to r + 35 = 35. This equation is also true, so the point lies on the graph of the second equation. Therefore, option A is a valid point for both equations.
2. Option B: Substituting the coordinates (-3r/2 + 7/2 , r) into the first equation, we get 2(-3r/2 + 7/2) +3(r) = 7, which simplifies to -3r + 7 + 3r = 7. This equation is true, so the point lies on the graph of the first equation. Substituting the same coordinates into the second equation, we get 10(-3r/2 + 7/2) +15(r) = 35, which simplifies to -15r + 35 + 15r = 35. This equation is also true, so the point lies on the graph of the second equation. Therefore, option B is a valid point for both equations.
3. ... and so on.

After evaluating each option, we can conclude that options A, B, and D all lie on the graph of each equation in the xy-plane for the given system.