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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineRob and Julia decided to shoot arrows at a simple target with a large outer ring and a smaller bull's-eye. Rob went first and landed 55 arrows in the outer ring and 55 arrows in the bull's-eye, for a total of 375375 points. Julia went second and got 55 arrows in the outer ring and 44 arrows in the bull's-eye, earning a total of 316316 points. How many points is each region of the target worth?\newlineThe outer ring is worth _\_ points, and the bull's-eye is worth _\_ points.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineRob and Julia decided to shoot arrows at a simple target with a large outer ring and a smaller bull's-eye. Rob went first and landed 55 arrows in the outer ring and 55 arrows in the bull's-eye, for a total of 375375 points. Julia went second and got 55 arrows in the outer ring and 44 arrows in the bull's-eye, earning a total of 316316 points. How many points is each region of the target worth?\newlineThe outer ring is worth _\_ points, and the bull's-eye is worth _\_ points.
  1. Define Variables: Define the variables for the points of each region of the target.\newlineLet xx represent the points for the outer ring.\newlineLet yy represent the points for the bull's-eye.
  2. Write Equations: Write the system of equations based on the information given.\newlineRob's score: 55 arrows in the outer ring and 55 arrows in the bull's-eye for a total of 375375 points.\newlineJulia's score: 55 arrows in the outer ring and 44 arrows in the bull's-eye for a total of 316316 points.\newlineThe system of equations is:\newline5x+5y=3755x + 5y = 375 (Rob's score)\newline5x+4y=3165x + 4y = 316 (Julia's score)
  3. Use Elimination: Use elimination to solve the system of equations.\newlineWe can eliminate xx by subtracting the second equation from the first equation.\newline(5x+5y)−(5x+4y)=375−316(5x + 5y) - (5x + 4y) = 375 - 316\newline5x+5y−5x−4y=375−3165x + 5y - 5x - 4y = 375 - 316\newliney=59y = 59
  4. Substitute Values: Substitute the value of yy into one of the original equations to solve for xx. Using Rob's score equation: 5x+5(59)=3755x + 5(59) = 375 5x+295=3755x + 295 = 375 5x=375−2955x = 375 - 295 5x=805x = 80 x=16x = 16
  5. Verify Solution: Verify the solution by substituting the values of xx and yy into the second equation.\newlineUsing Julia's score equation:\newline5(16)+4(59)=3165(16) + 4(59) = 316\newline80+236=31680 + 236 = 316\newline316=316316 = 316\newlineThe values satisfy the second equation.

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