Problem
Solve the following system of equations by graphing. If the system is inconsistent or the equations are dependent, say so. \[ \begin{array}{r} x+2 y=10 \\ 2 x+4 y=32 \end{array} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The system has a single solution. The solution set is \{\} (Type an ordered pair) B. There are infinitely many solutions and the equations are dependent. The solution set is $\{(x, y) \mid x+2 y=10\}$. C. The system is inconsistent. The solution set is the empty set.
Solution
Step 1 ：The given system of equations is: \(x + 2y = 10\) and \(2x + 4y = 32\)
Step 2 ：We can simplify the second equation by dividing every term by 2, which gives us: \(x + 2y = 16\)
Step 3 ：Now, we have two equations: \(x + 2y = 10\) and \(x + 2y = 16\)
Step 4 ：These two equations are parallel and do not intersect, which means there are no solutions. Therefore, the system is inconsistent.
Step 5 ：\(\boxed{\text{The system is inconsistent. The solution set is the empty set.}}\)
