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Write a system of equations to describe the situation below, solve using an augmented matrix, 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 88 round tables and 55 rectangular tables, which will require a total of 3131 centerpieces. On the left side, there will be 11 rectangular table, for which they will need to assemble a total of 33 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 an augmented matrix, 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 88 round tables and 55 rectangular tables, which will require a total of 3131 centerpieces. On the left side, there will be 11 rectangular table, for which they will need to assemble a total of 33 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 xx be the number of centerpieces on each round table and yy be the number of centerpieces on each rectangular table.
  2. Equation for Right Side: For the right side of the room: 8x8x (round tables) + 5y5y (rectangular tables) = 3131 centerpieces.
  3. Equation for Left Side: For the left side of the room: 1y1y (rectangular table) = 33 centerpieces.
  4. System of Equations: We have the system of equations:\newline8x+5y=318x + 5y = 31\newline0x+1y=30x + 1y = 3
  5. Convert to Augmented Matrix: Convert the system of equations into an augmented matrix:\newline\begin{array}{cc|c} 8 & 5 & 31 \ 0 & 1 & 3 \end{array}
  6. Solve for yy: We can see that the second equation already gives us the value of yy, so y=3y = 3.
  7. Substitute yy into First Equation: Substitute y=3y = 3 into the first equation: 8x+5(3)=318x + 5(3) = 31.
  8. Calculate for x: Calculate: 8x+15=318x + 15 = 31.
  9. Solve for x: Subtract 1515 from both sides: 8x=31−158x = 31 - 15.
  10. Solve for x: Subtract 1515 from both sides: 8x=31−158x = 31 - 15. Calculate: 8x=168x = 16.
  11. Solve for x: Subtract 1515 from both sides: 8x=31−158x = 31 - 15. Calculate: 8x=168x = 16. Divide both sides by 88: x=168x = \frac{16}{8}.
  12. Solve for x: Subtract 1515 from both sides: 8x=31−158x = 31 - 15.Calculate: 8x=168x = 16.Divide both sides by 88: x=16/8x = 16 / 8.Calculate: x=2x = 2.

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