Write a system of equations to describe the situation below, solve using elimination, 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 $3$ small candle holders and $7$ large candle holders, using a total of $65$ candles. On the west side, he replaced the candles in $20$ small candle holders and $7$ large candle holders, for a total of $116$ 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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Algebra 1

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 elimination, 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

3

small candle holders and

7

large candle holders, using a total of

65

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

20

small candle holders and

7

large candle holders, for a total of

116

candles. How many candles does each candle holder hold?

\newline

Each small candleholder holds

\_

candles, and each large one holds

\_

candles.

Define variables: Define the variables for the number of candles each size of candle holder holds. $\newline$ Let $x$ be the number of candles a small candle holder holds. $\newline$ Let $y$ be the number of candles a large candle holder holds.
Write equations: Write the system of equations based on the given information. $\newline$ For the east side: $3x + 7y = 65$ $\newline$ For the west side: $20x + 7y = 116$
Eliminate variable: Decide which variable to eliminate. $\newline$ We can eliminate $y$ because it has the same coefficient in both equations.
Subtract equations: Subtract the first equation from the second to eliminate $y$ . $\newline$ $(20x + 7y) - (3x + 7y) = 116 - 65$ $\newline$ $20x + 7y - 3x - 7y = 116 - 65$ $\newline$ $17x = 51$
Solve for x: Solve for x. $\newline$ $17x = 51$ $\newline$ $x = \frac{51}{17}$ $\newline$ $x = 3$
Substitute $x$ : Substitute $x$ back into one of the original equations to solve for $y$ . Using the first equation: $3x + 7y = 65$ $3(3) + 7y = 65$ $9 + 7y = 65$ $7y = 65 - 9$ $7y = 56$ $y = \frac{56}{7}$ $y = 8$
Final answer: Write the final answer. $\newline$ Each small candleholder holds $3$ candles, and each large one holds $8$ candles.