Solve the non-linear system of equations. \( \begin{array}{l}(x-11)^{2}+(y)^{2}=25 \\ -2 x+y=-17\end{array} \) Present your answer in the form of \( (x, y) \). If there are more than one solutions, separate them by a comma. If there is no solution, enter DNE.
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To solve the non-linear system of equations, we begin with: 1. \((x-11)^{2}+(y)^{2}=25\) (This represents a circle centered at (11, 0) with a radius of 5) 2. \(-2x+y=-17\) (This can be rewritten as \(y=2x-17\)) Next, we substitute \(y\) from the second equation into the first equation: \((x-11)^{2} + (2x-17)^{2} = 25\) Expanding both sides: 1. \((x-11)^{2} = x^2 - 22x + 121\) 2. \((2x-17)^{2} = 4x^{2} - 68x + 289\) Now substitute these expansions into the equation: \[ x^2 - 22x + 121 + 4x^{2} - 68x + 289 = 25 \] Combine like terms: \[ 5x^{2} - 90x + 410 = 25 \] Subtract 25 from both sides: \[ 5x^{2} - 90x + 385 = 0 \] Now we divide everything by 5: \[ x^{2} - 18x + 77 = 0 \] Next, we can solve this quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\): Here, \(a = 1,\ b = -18,\ c = 77\) Calculating the discriminant: \[ D = b^2 - 4ac = (-18)^{2} - 4(1)(77) = 324 - 308 = 16 \] Since the discriminant is positive, there are two solutions: \[ x = \frac{18 \pm \sqrt{16}}{2(1)} \] So, \[ x = \frac{18 \pm 4}{2} \] This gives us: 1. \(x = \frac{22}{2} = 11\) 2. \(x = \frac{14}{2} = 7\) Next, substitute these \(x\) values back to find \(y\): 1. For \(x = 11\): \[ y = 2(11) - 17 = 22 - 17 = 5 \Rightarrow (11, 5) \] 2. For \(x = 7\): \[ y = 2(7) - 17 = 14 - 17 = -3 \Rightarrow (7, -3) \] Thus, the solutions to the non-linear system of equations are: **(11, 5), (7, -3)**
