Question: Solve the system by elimination. If the system is inconsistent or has dependent equations, say so…

Solve the system by elimination. If the system is inconsistent or has dependent equations, say so.

9x + 9y = 50 x + y = 5

Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice.

A. The system has a single solution. The solution set is (Type an ordered pair. Type integers or simplified fractions.)

B. There are infinitely many solutions and the equations are dependent. The solution set is {(x, y) | 9x + 9y = 50}

C. The system is inconsistent. The solution set is the empty set.

Solution

To solve the system by elimination, we’ll start with the given equations: \[ \begin{cases} 9x + 9y = 50 & \\ x + y = 5 & \end{cases} \] First, let’s eliminate \(y\) by multiplying the second equation by 9 to align it with the first: \[ 9(x + y) = 9 \times 5 \] This gives: \[ 9x + 9y = 45 \] Now we have: \[ \begin{cases} 9x + 9y = 50 & \\ 9x + 9y = 45 & \end{cases} \] Subtract the second equation from the first: \[ (9x + 9y) - (9x + 9y) = 50 - 45 \] This simplifies to: \[ 0 = 5 \] Since the resultant equation is a contradiction (0 ≠ 5), the system is inconsistent. Therefore, the correct choice is: C. The system is inconsistent. The solution set is the empty set.