\begin{tabular}{|l|l} Solve by the elimination method. \\ \( \begin{array}{l}5 x-y=9 \\ x+6 y=8\end{array} \) & \( \begin{array}{l}\text { Select the correct choice below and, if necessary, fill in the answer box to complete your choice. } \\ \text { (Simplify your answer. Type an ordered pair.) } \\ \text { ( B. There are infinitely many solutions. } \\ \text { OC. There is no solution. }\end{array} \)\end{tabular}
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To solve the system of equations by the elimination method, we first write the equations clearly: 1. \(5x - y = 9\) (Equation 1) 2. \(x + 6y = 8\) (Equation 2) To eliminate \(y\), we can multiply Equation 1 by 6 to match the coefficient of \(y\) in Equation 2: \[ 30x - 6y = 54 \quad \text{(Equation 1 multiplied by 6)} \] Now we have: 1. \(30x - 6y = 54\) 2. \(x + 6y = 8\) Next, we add the two equations together: \[ (30x - 6y) + (x + 6y) = 54 + 8 \] This simplifies to: \[ 31x = 62 \] Now, solve for \(x\): \[ x = 2 \] Now that we have \(x\), we can substitute \(x = 2\) back into one of the original equations to solve for \(y\). Let's use Equation 1: \[ 5(2) - y = 9 \] This simplifies to: \[ 10 - y = 9 \] Rearranging gives: \[ -y = -1 \quad \Rightarrow \quad y = 1 \] So the solution to the system of equations is: \[ (x, y) = (2, 1) \] Thus, the ordered pair is \( \boxed{(2, 1)} \). Since we found a unique solution, the answer is not infinitely many or no solution.
