Solve the system of equations. \[ \begin{array}{l}y=x^{2}-29 x-49 \\ y=-29 x+72\end{array} \] Write the coordinates in exact form. Simplify all fractions and ra
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To solve the system of equations, we can set the two equations equal to each other since both equal \(y\): \[ x^{2} - 29x - 49 = -29x + 72 \] Now we can simplify this by adding \(29x\) to both sides: \[ x^{2} - 49 = 72 \] Next, add \(49\) to both sides: \[ x^{2} = 121 \] Taking the square root of both sides, we find: \[ x = 11 \quad \text{or} \quad x = -11 \] Now, we'll substitute these values of \(x\) back into one of the original equations to find the corresponding \(y\) values. Let's use the second equation \(y = -29x + 72\). 1. For \(x = 11\): \[ y = -29(11) + 72 = -319 + 72 = -247 \] So one solution is \((11, -247)\). 2. For \(x = -11\): \[ y = -29(-11) + 72 = 319 + 72 = 391 \] So the second solution is \((-11, 391)\). Thus, the coordinates in exact form are: \[ (11, -247) \quad \text{and} \quad (-11, 391) \]
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