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Solve the system of equations.\newliney=−27x+49y = -27x + 49\newliney=x2−16x+7y = x^2 - 16x + 7\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)

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Solve a nonlinear system of equationsright chevron icon

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Q. Solve the system of equations.\newliney=−27x+49y = -27x + 49\newliney=x2−16x+7y = x^2 - 16x + 7\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)
  1. Set Equations Equal: Set the two equations equal to each other since they both equal yy. This gives us −27x+49=x2−16x+7-27x + 49 = x^2 - 16x + 7.
  2. Rearrange and Form Quadratic: Rearrange the equation to set it to zero and form a quadratic equation. This gives us x2−16x+27x+7−49=0x^2 - 16x + 27x + 7 - 49 = 0, which simplifies to x2+11x−42=0x^2 + 11x - 42 = 0.
  3. Factor Quadratic Equation: Factor the quadratic equation. We are looking for two numbers that multiply to −42-42 and add up to 1111. These numbers are 1414 and −3-3. So the factored form is (x+14)(x−3)=0(x + 14)(x - 3) = 0.
  4. Solve for x: Solve for x by setting each factor equal to zero. This gives us x+14=0x + 14 = 0 or x−3=0x - 3 = 0. Solving these gives us x=−14x = -14 and x=3x = 3.
  5. Substitute xx Values: Substitute x=−14x = -14 into one of the original equations to find the corresponding yy value. Using y=−27x+49y = -27x + 49 gives us y=−27(−14)+49y = -27(-14) + 49, which simplifies to y=378+49y = 378 + 49 and then y=427y = 427.
  6. Find Corresponding y Values: Substitute x=3x = 3 into the same original equation to find the corresponding y value. Using y=−27x+49y = -27x + 49 gives us y=−27(3)+49y = -27(3) + 49, which simplifies to y=−81+49y = -81 + 49 and then y=−32y = -32.

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