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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineFriends and family of the bride will be helping assemble centerpieces for the wedding reception. On the right side of the room, there will be 11 round table and 33 rectangular tables, which will require a total of 1010 centerpieces. On the left side, there will be 1515 round tables and 22 rectangular tables, for which they will need to assemble a total of 2121 centerpieces. How many centerpieces will be on each table?\newlineThere will be _____ centerpieces on every round table and _____ centerpieces on every rectangular one.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineFriends and family of the bride will be helping assemble centerpieces for the wedding reception. On the right side of the room, there will be 11 round table and 33 rectangular tables, which will require a total of 1010 centerpieces. On the left side, there will be 1515 round tables and 22 rectangular tables, for which they will need to assemble a total of 2121 centerpieces. How many centerpieces will be on each table?\newlineThere will be _____ centerpieces on every round table and _____ centerpieces on every rectangular one.
  1. Define Variables: Let's denote the number of centerpieces on each round table as rr and the number of centerpieces on each rectangular table as tt. On the right side of the room, there is 11 round table and 33 rectangular tables, requiring a total of 1010 centerpieces. This gives us the equation 1r+3t=101r + 3t = 10.
  2. Form Equations: On the left side of the room, there are 1515 round tables and 22 rectangular tables, needing a total of 2121 centerpieces. This gives us the equation 15r+2t=2115r + 2t = 21.
  3. Eliminate Variable: We now have a system of two equations. We need to eliminate one of the variables, rr or tt. We choose to eliminate tt because its coefficients are smaller and easier to work with.
  4. Correct Subtraction: To eliminate tt, we multiply the first equation by 22, the coefficient of tt in the second equation. This gives us the new equation 2r+6t=202r + 6t = 20.
  5. Correct Subtraction: To eliminate tt, we multiply the first equation by 22, the coefficient of tt in the second equation. This gives us the new equation 2r+6t=202r + 6t = 20.We now subtract the new first equation from the second equation to eliminate tt. This gives us 15r−2r+2t−6t=21−2015r - 2r + 2t - 6t = 21 - 20, which simplifies to 13r−4t=113r - 4t = 1.
  6. Correct Subtraction: To eliminate tt, we multiply the first equation by 22, the coefficient of tt in the second equation. This gives us the new equation 2r+6t=202r + 6t = 20.We now subtract the new first equation from the second equation to eliminate tt. This gives us 15r−2r+2t−6t=21−2015r - 2r + 2t - 6t = 21 - 20, which simplifies to 13r−4t=113r - 4t = 1.We made a mistake in the previous step by not correctly subtracting the equations. Let's correct this. Subtracting the new first equation from the second equation should give us 15r+2t−(2r+6t)=21−2015r + 2t - (2r + 6t) = 21 - 20, which simplifies to 13r−4t=113r - 4t = 1. However, we should have subtracted the entire first equation from the second equation, not just the rr terms. Let's correct this and subtract the equations properly.

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