Problem
Solve the system. State whether it is inconsistent or has infinitely many solutions. If the system has infinitely $1 \leftarrow \quad$ many solutions, write the solution set with y arbitrary. \[ \begin{array}{r} 7 x-7 y-5=0 \\ x-y-12=0 \end{array} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There is one solution. The solution set is (Simplify your answer. Type an ordered pair.) B. The system has infinitely many solutions: The solution set is \{ (Simplify your answer. Type an ordered pair. Type an expression using y as the variable.) C. The solution is inconsistent.
Solution
Step 1 ：The system of equations is given by: \(7x - 7y - 5 = 0\) and \(x - y - 12 = 0\)
Step 2 ：We can simplify the first equation by dividing all terms by 7, which gives us: \(x - y - \frac{5}{7} = 0\)
Step 3 ：Now we have two equations with the same structure, which are: \(x - y - \frac{5}{7} = 0\) and \(x - y - 12 = 0\)
Step 4 ：We can see that the left-hand side of both equations is the same (\(x - y\)), but the right-hand side is different (-\(\frac{5}{7}\) and -12). This means that the system of equations is inconsistent, because there is no pair (\(x, y\)) that can satisfy both equations at the same time.
Step 5 ：\(\boxed{\text{The solution is inconsistent.}}\)
