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
what is 3(4x+6)+7x
To determine which of the linear systems has exactly one solution, we can use a method called the elimination method or the substitution method. Let's go through each linear system and check if it has one solution.
A: -x+y=9
x-y=9
We can eliminate one variable by adding the two equations together. Adding the left sides gives us -x + x + y - y = 9 + 9, which simplifies to 0 = 18. This means that the variables cancel out and we are left with 0 = 18, which is not a true statement. Since this equation is contradictory, the linear system does not have a solution.
B: -x+y=9
x-y= -9
Again, let's eliminate one variable by adding the two equations together. Adding the left sides gives us -x + x + y - y = 9 + (-9), which simplifies to 0 = 0. This is a true statement, indicating that the two equations represent the same line. When two lines coincide, they have infinitely many solutions, not just one. Therefore, this linear system does not have exactly one solution.
C: -x+y=9
-x-y=9
Let's add the two equations together to eliminate a variable. Adding the left sides gives us -x + (-x) + y - y = 9 + 9, which simplifies to -2x = 18. Dividing both sides of this equation by -2 gives us x = -9. Now, substituting this value of x back into one of the equations, say the first equation, we get -(-9) + y = 9, which simplifies to 9 + y = 9. By subtracting 9 from both sides, we get y = 0. Therefore, this linear system has a unique solution of x = -9 and y = 0, which means it has exactly one solution.
D: x-y= -9
-x+y= -9
Let's add the two equations together to eliminate a variable. Adding the left sides gives us x + (-x) - y + y = -9 + (-9), which simplifies to 0 = -18. This is not a true statement, so the linear system does not have a solution.
In conclusion, out of the given linear systems, only option C: -x+y=9, -x-y=9 has exactly one solution.