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
find a system where the slopes are different. ANy two lines with different slopes intersect at exactly one point.
Looks like (C)
|1.5C-24|<=30
To determine which of the linear systems has exactly one solution, we need to find the number of solutions for each system.
To do this, we can use various methods such as substitution, elimination, or graphing. Let's use the method of elimination for each of the given linear systems:
A: -x+y=9
x-y=9
Adding the two equations, we eliminate the y term:
( -x+y ) + ( x-y ) = 9 + 9
-2y = 18
y = -9
Substituting the value of y into one of the original equations:
-x - 9 = 9
-x = 9 + 9
-x = 18
x = -18
So in this case, the solution is x = -18, y = -9.
B: -x+y=9
x-y= -9
Adding the two equations, we eliminate the y term:
( -x+y ) + ( x-y ) = 9 + (-9)
0 = 0
This equation is always true, meaning there are infinitely many solutions.
C: -x+y=9
-x-y=9
Adding the two equations, we eliminate the x term:
( -x+y ) + ( -x-y ) = 9 + 9
-2x = 18
x = -9
Substituting the value of x into one of the original equations:
-(-9) + y = 9
9 + y = 9
y = 0
So in this case, the solution is x = -9, y = 0.
D: x-y= -9
-x+y= -9
Adding the two equations, we eliminate both the x and y terms:
( x-y ) + ( -x+y ) = -9 + (-9)
0 = -18
This equation is never true, meaning there are no solutions.
Based on our analysis, option A (-x+y=9, x-y=9) is the only linear system that has exactly one solution, which is x = -18, y = -9.