Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ At a historical landmark, candles are used to simulate an authentic atmosphere. A volunteer is currently putting new candles in the candle holders. On the east side, he replaced candles in $9$ small candle holders and $11$ large candle holders, using a total of $93$ candles. On the west side, he replaced the candles in $12$ small candle holders and $11$ large candle holders, for a total of $102$ candles. How many candles does each candle holder hold? $\newline$ Each small candleholder holds $\_$ candles, and each large one holds $\_$ candles.

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Solve a system of equations using any method: word problems

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

At a historical landmark, candles are used to simulate an authentic atmosphere. A volunteer is currently putting new candles in the candle holders. On the east side, he replaced candles in

9

small candle holders and

11

large candle holders, using a total of

93

candles. On the west side, he replaced the candles in

12

small candle holders and

11

large candle holders, for a total of

102

candles. How many candles does each candle holder hold?

\newline

Each small candleholder holds

\_

candles, and each large one holds

\_

candles.

Equation $1$ : Let's denote the number of candles a small candle holder holds as $x$ and the number of candles a large candle holder holds as $y$ . The volunteer replaced candles in $9$ small candle holders and $11$ large candle holders on the east side, using a total of $93$ candles. This gives us our first equation: $\newline$ $9x + 11y = 93$
Equation $2$ : On the west side, the volunteer replaced the candles in $12$ small candle holders and $11$ large candle holders, for a total of $102$ candles. This gives us our second equation: $\newline$ $12x + 11y = 102$
Solving System: We now have a system of equations: $\newline$ $9x + 11y = 93$ $\newline$ $12x + 11y = 102$ $\newline$ To solve the system, we can subtract the first equation from the second to eliminate $y$ . $\newline$ $(12x + 11y) - (9x + 11y) = 102 - 93$ $\newline$ $12x + 11y - 9x - 11y = 9$ $\newline$ $3x = 9$ $\newline$ $x = 3$
Substitute $x$ : Now that we have the value for $x$ , we can substitute it back into one of the original equations to solve for $y$ . Let's use the first equation: $\newline$ $9(3) + 11y = 93$ $\newline$ $27 + 11y = 93$ $\newline$ $11y = 93 - 27$ $\newline$ $11y = 66$ $\newline$ $y = 6$