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
4 - y = 10
-y=10-4=6
To determine which of the linear systems has exactly one solution, we need to analyze each system separately and check the number of solutions it has.
System A:
-x + y = 9
x - y = 9
To find the solution to this system, we can add the two equations together:
(-x + y) + (x - y) = 9 + 9
This simplifies to:
0 = 18
The equation 0 = 18 is not true, which means there are no solutions to this system. Therefore, System A does not have exactly one solution.
System B:
-x + y = 9
x - y = -9
Similarly, we can add the two equations together:
(-x + y) + (x - y) = 9 + (-9)
This simplifies to:
0 = 0
The equation 0 = 0 holds true. This means that the two equations in System B represent the same line and intersect at every point along that line. Therefore, System B has infinitely many solutions and does not have exactly one solution.
System C:
-x + y = 9
-x - y = 9
Again, we can add the two equations together:
(-x + y) + (-x - y) =9 + 9
This simplifies to:
-2x = 18
Dividing both sides of the equation by -2 gives us:
x = -9
By substituting this value back into one of the original equations, we can solve for y:
-x + y = 9
-(-9) + y = 9
9 + y = 9
y = 0
The solution to System C is x = -9 and y = 0. Since there is a unique solution to this system, System C has exactly one solution.
System D:
x - y = -9
-x + y = -9
Adding the two equations together yields:
(x - y) + (-x + y) = -9 + (-9)
This simplifies to:
0 = -18
The equation 0 = -18 is not true, so System D has no solutions and does not have exactly one solution.
In conclusion, out of the given linear systems, only System C (-x + y = 9 and -x - y = 9) has exactly one solution.