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
Start putting all the equations into the same form y = a*x + b
A: -x+y=9; y =x + 9
x-y=9; x = 9 + y; y =x-9
These two are not the same
B: -x+y=9; y = 9+x
x-y= -9; x = -9+y; y = x+9
These two are the same
C: -x+y=9; y = 9+x
-x-y=9; -y = 9+x; y = -x -9
These two are not the same
D: x-y= -9; x+9 = y
-x+y= -9 ; y = x-9
These two are not the same
The correct answer is B
To determine which of the linear systems has exactly one solution, we can use the concept of linear independence.
If a system of equations has exactly one solution, it means that the two equations are not dependent on each other and that they intersect at a single point.
Let's examine each option:
A: -x+y=9, x-y=9
To solve this system, we can multiply the second equation by -1 to eliminate the y term:
-x + y = 9
x - y = 9
-------------------
2x = 18
Dividing both sides by 2, we get x = 9.
Plugging this value back into either equation, we find that y = 0.
Thus, option A has a unique solution (9, 0).
B: -x+y=9, x-y=-9
Following the same steps as before, we eliminate the y term by multiplying the second equation by -1:
-x + y = 9
-x + y = 9
----------------
0 = 0
In this case, the two equations are identical, meaning they represent the same line. Therefore, there are infinitely many solutions, not exactly one.
So, option B does not have exactly one solution.
C: -x+y=9, -x-y=9
By adding the two equations together, we get:
-x + y = 9
+ -x - y = 9
----------------
-2x = 18
Dividing both sides by -2, we get x = -9.
Plugging this value back into either equation, we find that y = -18.
Thus, option C has a unique solution (-9, -18).
D: x - y = -9, -x + y = -9
Adding the two equations together, we get:
x - y = -9
+ -x + y = -9
----------------
0 = -18
In this case, the two equations are contradictory, resulting in an inconsistent system. There are no values that satisfy both equations, meaning there is no solution.
So, option D does not have exactly one solution.
In summary, the linear system that has exactly one solution is option A: -x + y = 9, x - y = 9.