Question: Fill in the blank to correctly complete the sentence. The following nonlinear system has two…

Fill in the blank to correctly complete the sentence.

The following nonlinear system has two solutions, one of which is $(6, \\)$.

\[ x + y = 13 \]

\[ x^2 + y^2 = 85 \]

One of the solutions of the system is $(6, \\)$.

(Type an integer or a simplified fraction.)

Solution

We want to solve the following nonlinear system with the condition that one of the solutions is \((6, y)\): \[ \begin{cases} x + y = 13 & \\ x^2 + y^2 = 85 & \end{cases} \] Substitute \(x = 6\) into the first equation: \[ 6 + y = 13 \] Solve for \(y\): \[ y = 13 - 6 \] \[ y = 7 \] Now verify the solution by substituting \(x = 6\) and \(y = 7\) into the second equation: \[ 6^2 + 7^2 = 85 \] \[ 36 + 49 = 85 \] \[ 85 = 85 \] This confirms that \((6, 7)\) is indeed a solution. The solution is \((6, 7)\).