Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this.

9x + 6y = 60

-27x Ð 18y = -180

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Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this.

9x + 6y = 60

-27x Ð 18y = -180

To solve the system of equations using the elimination method, we will multiply the equations by appropriate coefficients to create oppositely signed terms for one of the variables. Then, we will add or subtract the equations to eliminate that variable and solve for the remaining variable.

Given the system of equations:

Equation 1: 9x + 6y = 60
Equation 2: -27x - 18y = -180

To eliminate the x variable, we need to multiply one equation by a number such that when added or subtracted, the x terms will cancel.

Let's multiply Equation 1 by 3 and Equation 2 by -1:

3(9x + 6y) = 3(60) => 27x + 18y = 180 (Equation 3)
-1(-27x - 18y) = -1(-180) => 27x + 18y = 180 (Equation 4)

Now, we can notice that Equation 3 and Equation 4 have exactly the same x and y terms. When we subtract Equation 4 from Equation 3, the x terms will cancel out:

(27x + 18y) - (27x + 18y) = 180 - 180
0 = 0

This equation indicates that 0 = 0, which is true. This means that the variable y has been eliminated from the equations.

However, since there is no equation left to solve for x, we cannot find a unique solution for this system of equations. This implies that the system has infinitely many solutions or is an inconsistent system.

To summarize:
The system of equations has no unique solution. It either has infinitely many solutions or is inconsistent.