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 analyze each system of equations. Each system contains two linear equations in two variables (x and y).
A: -x + y = 9
x - y = 9
We can solve this system by adding the two equations together:
(-x + y) + (x - y) = 9 + 9
0 = 18
The result is a contradiction (0 = 18), which means the system has no solution. Therefore, System A does not have exactly one solution.
B: -x + y = 9
x - y = -9
We can again add the two equations together:
(-x + y) + (x - y) = 9 + (-9)
0 = 0
The result is a consistent and dependent system, which means that there are infinitely many solutions. Therefore, System B does not have exactly one solution.
C: -x + y = 9
-x - y = 9
We can add the two equations together:
(-x + y) + (-x - y) = 9 + 9
-2x = 18
x = -9
Substituting the value of x into either equation, we find:
-(-9) + y = 9
9 + y = 9
y = 0
So, the solution to this system is x = -9 and y = 0. Since we have determined a specific unique solution, System C has exactly one solution.
D: x - y = -9
-x + y = -9
We can add the two equations together:
(x - y) + (-x + y) = -9 + (-9)
0 = -18
The result is a contradiction (0 = -18), indicating that the system has no solution. Therefore, System D does not have exactly one solution.
In conclusion, only System C (-x + y = 9 and -x - y = 9) has exactly one solution.