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Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ Friends and family of the bride will be helping assemble centerpieces for the wedding reception. On the right side of the room, there will be $8$ round tables and $5$ rectangular tables, which will require a total of $31$ centerpieces. On the left side, there will be $1$ rectangular table, for which they will need to assemble a total of $3$ centerpieces. How many centerpieces will be on each table? $\newline$ There will be $\_\_\_\_$ centerpieces on every round table and $\_\_\_\_$ centerpieces on every rectangular one.

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Solve a system of equations using elimination: word problems

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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.

\newline

Friends and family of the bride will be helping assemble centerpieces for the wedding reception. On the right side of the room, there will be

8

round tables and

5

rectangular tables, which will require a total of

31

centerpieces. On the left side, there will be

1

rectangular table, for which they will need to assemble a total of

3

centerpieces. How many centerpieces will be on each table?

\newline

There will be

\_\_\_\_

centerpieces on every round table and

\_\_\_\_

centerpieces on every rectangular one.

Define Variables: Let $x$ be the number of centerpieces on each round table and $y$ be the number of centerpieces on each rectangular table.
Equation for Right Side: For the right side of the room: $8x$ (round tables) + $5y$ (rectangular tables) = $31$ centerpieces.
Equation for Left Side: For the left side of the room: $1y$ (rectangular table) = $3$ centerpieces.
System of Equations: We have the system of equations: $\newline$ $8x + 5y = 31$ $\newline$ $0x + 1y = 3$
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}
Solve for $y$ : We can see that the second equation already gives us the value of $y$ , so $y = 3$ .
Substitute $y$ into First Equation: Substitute $y = 3$ into the first equation: $8x + 5(3) = 31$ .
Calculate for x: Calculate: $8x + 15 = 31$ .
Solve for x: Subtract $15$ from both sides: $8x = 31 - 15$ .
Solve for x: Subtract $15$ from both sides: $8x = 31 - 15$ . Calculate: $8x = 16$ .
Solve for x: Subtract $15$ from both sides: $8x = 31 - 15$ . Calculate: $8x = 16$ . Divide both sides by $8$ : $x = \frac{16}{8}$ .
Solve for x: Subtract $15$ from both sides: $8x = 31 - 15$ .Calculate: $8x = 16$ .Divide both sides by $8$ : $x = 16 / 8$ .Calculate: $x = 2$ .

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