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{:[y=10+16 x-x^(2)],[y=3x+50]:} If (x_(1),y_(1)) and (x_(2),y_(2)) are distinct solutions to the system of equations shown, what is the sum of the y_(1) and y_(2) ?

y=10+16x−x2y=3x+50 \begin{array}{l} y=10+16 x-x^{2} \\ y=3 x+50 \end{array} \newlineIf (x1,y1) \left(x_{1}, y_{1}\right) and (x2,y2) \left(x_{2}, y_{2}\right) are distinct solutions to the system of equations shown, what is the sum of the y1 y_{1} and y2 y_{2} ?

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Q. y=10+16x−x2y=3x+50 \begin{array}{l} y=10+16 x-x^{2} \\ y=3 x+50 \end{array} \newlineIf (x1,y1) \left(x_{1}, y_{1}\right) and (x2,y2) \left(x_{2}, y_{2}\right) are distinct solutions to the system of equations shown, what is the sum of the y1 y_{1} and y2 y_{2} ?
  1. Write Equations: Write down the system of equations.\newlineWe have the following system of equations:\newliney=10+16x−x2y = 10 + 16x - x^2\newliney=3x+50y = 3x + 50
  2. Set Equal: Set the two equations equal to each other to find the xx-values where the yy-values are the same (the intersection points).10+16x−x2=3x+5010 + 16x - x^2 = 3x + 50
  3. Rearrange & Solve: Rearrange the equation to set it to zero and solve for xx.x2−16x+3x−10+50=0x^2 - 16x + 3x - 10 + 50 = 0x2−13x+40=0x^2 - 13x + 40 = 0
  4. Factor Quadratic: Factor the quadratic equation to find the values of xx.(x−5)(x−8)=0(x - 5)(x - 8) = 0
  5. Solve for x: Solve for the x-values x1x_1 and x2x_2.x−5=0x - 5 = 0 or x−8=0x - 8 = 0x1=5x_1 = 5, x2=8x_2 = 8
  6. Substitute & Find y1y_1: Substitute x1x_1 and x2x_2 into either of the original equations to find y1y_1 and y2y_2. We can use the second equation y=3x+50y = 3x + 50 for simplicity.\newlineFor x1=5x_1 = 5:\newliney1=3(5)+50y_1 = 3(5) + 50\newliney1=15+50y_1 = 15 + 50\newliney1=65y_1 = 65
  7. Substitute & Find y2y_2: Substitute x2x_2 into the same equation to find y2y_2.\newlineFor x2=8x_2 = 8:\newliney2=3(8)+50y_2 = 3(8) + 50\newliney2=24+50y_2 = 24 + 50\newliney2=74y_2 = 74
  8. Find Sum: Add y1y_1 and y2y_2 to find the sum.\newlineSum = y1+y2y_1 + y_2\newlineSum = 65+7465 + 74\newlineSum = 139139

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