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Solve the system of equations.\newlinex2+y2=200x^2 + y^2 = 200\newlinex=−3y−20x = -3y - 20\newline\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)

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Q. Solve the system of equations.\newlinex2+y2=200x^2 + y^2 = 200\newlinex=−3y−20x = -3y - 20\newline\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)\newline(______,______)
  1. Substitute xx into first equation: Substitute xx from the second equation into the first equation.\newlinex2+y2=200x^2 + y^2 = 200\newlinex=−3y−20x = -3y - 20\newline(−3y−20)2+y2=200(-3y - 20)^2 + y^2 = 200
  2. Expand and simplify: Expand the squared term and simplify the equation.\newline(−3y−20)2+y2=200(-3y - 20)^2 + y^2 = 200\newline9y2+120y+400+y2=2009y^2 + 120y + 400 + y^2 = 200\newline10y2+120y+400=20010y^2 + 120y + 400 = 200
  3. Set equation to zero: Subtract 200200 from both sides to set the equation to zero.\newline10y2+120y+400−200=010y^2 + 120y + 400 - 200 = 0\newline10y2+120y+200=010y^2 + 120y + 200 = 0
  4. Divide and simplify: Divide the entire equation by 1010 to simplify.\newline10y210+120y10+20010=010\frac{10y^2}{10} + \frac{120y}{10} + \frac{200}{10} = \frac{0}{10}\newliney2+12y+20=0y^2 + 12y + 20 = 0
  5. Factor the equation: Factor the quadratic equation.\newline(y+10)(y+2)=0(y + 10)(y + 2) = 0
  6. Solve for y: Solve for y by setting each factor equal to zero.\newliney+10=0y + 10 = 0 or y+2=0y + 2 = 0\newliney=−10y = -10 or y=−2y = -2
  7. Find xx values: Substitute yy back into x=−3y−20x = -3y - 20 to find the corresponding xx values.\newlineFor y=−10y = -10: x=−3(−10)−20x = -3(-10) - 20\newlinex=30−20x = 30 - 20\newlinex=10x = 10
  8. Find xx values: Substitute yy back into x=−3y−20x = -3y - 20 to find the corresponding xx values.\newlineFor y=−2y = -2: x=−3(−2)−20x = -3(-2) - 20\newlinex=6−20x = 6 - 20\newlinex=−14x = -14
  9. Write coordinates: Write the coordinates in exact form.\newlineFirst Coordinate: (10,−10)(10, -10)\newlineSecond Coordinate: (−14,−2)(-14, -2)

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