The point (2,5) is a solution to which system of equations? A) y=x-8 and 2x+y=7 B) y=-1/2x+6 and y=3x-1 C) y=x+2 and y=x+5 D) y=2/3x+6 and 3t+6x-18=0
To determine which system of equations the point (2,5) is a solution to, substitute the coordinates (x=2, y=5) into each equation and see if they are satisfied.
A) y = x - 8
Substituting the coordinates (x=2, y=5) into the equation, we have:
5 = 2 - 8
This equation is not true, so point (2,5) is not a solution to equation A.
B) y = -1/2x + 6 and y = 3x - 1
Substituting the coordinates (x=2, y=5) into the first equation, we have:
5 = -1/2 * 2 + 6
5 = -1 + 6
5 = 5
This equation is true.
Substituting the coordinates (x=2, y=5) into the second equation, we have:
5 = 3 * 2 - 1
5 = 6 - 1
5 = 5
This equation is true as well.
Therefore, point (2,5) is a solution to equation B.
C) y = x + 2 and y = x + 5
Substituting the coordinates (x=2, y=5) into the first equation, we have:
5 = 2 + 2
5 = 4
This equation is not true, so point (2,5) is not a solution to equation C.
D) y = 2/3x + 6 and 3t + 6x - 18 = 0
Substituting the coordinates (x=2, y=5) into the first equation, we have:
5 = 2/3 * 2 + 6
5 = 4/3 + 6
5 = 22/3
This equation is not true, so point (2,5) is not a solution to equation D.
Based on the above analysis, the point (2,5) is a solution to system B) y = -1/2x + 6 and y = 3x - 1.
To determine which system of equations the point (2,5) is a solution to, we can substitute the x and y values of the point into each equation and see which set of equations results in both equations being true.
For option A), substituting x = 2 and y = 5 into the equations gives:
y = x - 8 → 5 = 2 - 8 → 5 = -6 (not true)
2x + y = 7 → 2(2) + 5 = 7 → 4 + 5 = 7 → 9 = 7 (not true)
Therefore, (2,5) is not a solution for option A).
For option B), substituting x = 2 and y = 5 into the equations gives:
y = -1/2x + 6 → 5 = -1/2(2) + 6 → 5 = -1 + 6 → 5 = 5 (true)
y = 3x - 1 → 5 = 3(2) - 1 → 5 = 6 - 1 → 5 = 5 (true)
Therefore, (2,5) is a solution for option B).
For option C), substituting x = 2 and y = 5 into the equations gives:
y = x + 2 → 5 = 2 + 2 → 5 = 4 (not true)
y = x + 5 → 5 = 2 + 5 → 5 = 7 (not true)
Therefore, (2,5) is not a solution for option C).
For option D), substituting x = 2 and y = 5 into the equations gives:
y = 2/3x + 6 → 5 = 2/3(2) + 6 → 5 = 4/3 + 6 → 5 = 22/3 (not true)
3t + 6x - 18 = 0 → 3t + 6(2) - 18 = 0 → 3t + 12 - 18 = 0 → 3t - 6 = 0 → 3(t - 2) = 0
Therefore, (2,5) is not a solution for option D).
Therefore, the point (2,5) is a solution to system B), y=-1/2x+6 and y=3x-1.
To determine which system of equations the point (2,5) is a solution to, we can substitute the values x=2 and y=5 into each system and see which one makes the equations true.
A) y=x-8 and 2x+y=7:
Substituting x=2 and y=5 into the first equation:
5 = 2 - 8
5 = -6
This is not true, so (2,5) is not a solution to this system.
B) y=-1/2x+6 and y=3x-1:
Substituting x=2 into both equations:
5 = -1/2(2) + 6
5 = -1 + 6
5 = 5
This is true, so (2,5) is a solution to this system.
C) y=x+2 and y=x+5:
Substituting x=2 into both equations:
5 = 2 + 2
5 = 4
This is not true, so (2,5) is not a solution to this system.
D) y=2/3x+6 and 3t+6x-18=0:
The second equation in this system contains a variable t, not y, so it does not have a solution that can be checked for the point (2,5). Therefore, (2,5) is not a solution to this system.
Therefore, the point (2,5) is a solution to the system of equations in option B) y=-1/2x+6 and y=3x-1.