Use elimination to solve the system of equations. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent.
\[
\begin{array}{l}
5 x+y=-12 \\
7 x-3 y=-30
\end{array}
\]
Final Answer: The system of equations is consistent and the equations are independent. The solution to the system of equations is \(\boxed{x = -3, y = 3}\).
Step 1 :Given the system of equations: \[\begin{array}{l} 5x+y=-12 \\ 7x-3y=-30 \end{array}\]
Step 2 :First, we make the coefficients of y the same in both equations. This can be achieved by multiplying the first equation by 3 and the second equation by 1.
Step 3 :Then, we subtract the second equation from the first to eliminate y.
Step 4 :We solve for x and find that \(x = -3\).
Step 5 :Substitute \(x = -3\) into one of the original equations to solve for y. We find that \(y = 3\).
Step 6 :Since we get a valid solution, the system is consistent.
Step 7 :We check if the equations are dependent or independent by checking if one equation can be obtained from the other by multiplying it with a constant.
Step 8 :Since one equation cannot be obtained from the other by multiplying it with a constant, the equations are independent.
Step 9 :Final Answer: The system of equations is consistent and the equations are independent. The solution to the system of equations is \(\boxed{x = -3, y = 3}\).