Which of the linear systems has exactly one solution
A: -x+y=9
x-y=9
B: -x+y=9
x-y= -9
C: -x+y=9
-x-y=9
D: x-y= -9
-x+y= -9
To determine which of the linear systems has exactly one solution, we need to check if the system is consistent and if the equations are not dependent or independent.
To do this, we can use various methods such as substitution, elimination, or graphing. Let's use the method of elimination to solve these systems and see which ones have exactly one solution.
A: -x + y = 9
x - y = 9
Add the two equations together:
-2x = 18
Divide both sides by -2:
x = -9
Substitute the value of x back into one of the original equations:
-(-9) + y = 9
9 + y = 9
y = 0
So the solution to system A is x = -9 and y = 0.
B: -x + y = 9
x - y = -9
Add the two equations together:
0 = 0
This implies that the system has infinitely many solutions because the equations are equivalent. Every point on the line -x + y = 9 satisfies both equations, so there is not exactly one solution for system B.
C: -x + y = 9
-x - y = 9
Add the two equations together:
-2x = 18
Divide both sides by -2:
x = -9
Substitute the value of x back into one of the original equations:
-(-9) + y = 9
9 + y = 9
y = 0
So the solution to system C is x = -9 and y = 0.
D: x - y = -9
-x + y = -9
Add the two equations together:
0 = -18
This implies that the system is inconsistent since the equations contradict each other. There is no solution for system D.
In summary, among the given systems, only systems A and C have exactly one solution.