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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 66 round tables and 22 rectangular tables, which will require a total of 1414 centerpieces. On the left side, there will be 66 round tables and 11 rectangular table, for which they will need to assemble a total of 1010 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 66 round tables and 22 rectangular tables, which will require a total of 1414 centerpieces. On the left side, there will be 66 round tables and 11 rectangular table, for which they will need to assemble a total of 1010 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 define variables for the number of centerpieces on each type of table.\newlineLet xx be the number of centerpieces on each round table.\newlineLet yy be the number of centerpieces on each rectangular table.\newlineWe can now write two equations based on the given information.
  2. Write equations for right side: Write the equation for the right side of the room.\newline66 round tables and 22 rectangular tables require 1414 centerpieces.\newlineThis gives us the equation: 6x+2y=146x + 2y = 14.
  3. Write equations for left side: Write the equation for the left side of the room.\newline66 round tables and 11 rectangular table require 1010 centerpieces.\newlineThis gives us the equation: 6x+y=106x + y = 10.
  4. Eliminate variable yy: We have the system of equations:\newline6x+2y=146x + 2y = 14\newline6x+y=106x + y = 10\newlineWe need to eliminate one of the variables. We can eliminate yy by multiplying the second equation by −2-2 and then adding it to the first equation.
  5. Multiply second equation by −2-2: Multiply the second equation by −2＂):$−2(6x+y)=−2(10)-2＂): \$-2(6x + y) = -2(10) −12x−2y=−20-12x - 2y = -20 Now we have the new system of equations: 6x+2y=146x + 2y = 14 −12x−2y=−20-12x - 2y = -20
  6. Add equations to eliminate y: Add the two equations to eliminate y:\newline(6x+2y)+(−12x−2y)=14+(−20)(6x + 2y) + (-12x - 2y) = 14 + (-20)\newline6x−12x+2y−2y=14−206x - 12x + 2y - 2y = 14 - 20\newline−6x=−6-6x = -6
  7. Solve for x: Solve for x:\newline−6x=−6-6x = -6\newlineDivide both sides by −6-6:\newlinex=1x = 1\newlineSo, there is 11 centerpiece on each round table.
  8. Substitute xx into second equation: Substitute x=1x = 1 into the second original equation to solve for yy:6x+y=106x + y = 106(1)+y=106(1) + y = 106+y=106 + y = 10y=10−6y = 10 - 6y=4y = 4So, there are 44 centerpieces on each rectangular table.

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