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 8 small candle holders and 6 large candle holders, using a total of 72 candles. On the west side, he replaced the candles in 18 small candle holders and 6 large candle holders, for a total of 102 candles. How many candles does a small candleholder and how many candles does a large candleholder hold?
Let's call the number of candles a small candle holder holds x and the number of candles a large candle holder holds y.
On the east side, the volunteer replaced candles in 8 small candle holders and 6 large candle holders, using a total of 8x + 6y = 72 candles.
Similarly, on the west side, the volunteer replaced candles in 18 small candle holders and 6 large candle holders, using a total of 18x + 6y = 102 candles.
We now have the following system of equations:
8x + 6y = 72
18x + 6y = 102
To solve this system, first, let's multiply the first equation by 3:
3 * (8x + 6y) = 3 * 72
24x + 18y = 216
Now, let's subtract the equation 18x + 6y = 102 from the equation 24x + 18y = 216:
24x + 18y - (18x + 6y) = 216 - 102
6x + 12y = 114
Simplifying the equation above, we get:
x + 2y = 19
Now, we need to solve the following system of equations:
x + 2y = 19
18x + 6y = 102
To solve it, let's multiply the first equation by 18:
18 * (x + 2y) = 18 * 19
18x + 36y = 342
Now, let's subtract the equation 18x + 36y = 342 from the equation 18x + 6y = 102:
(18x + 36y) - (18x + 6y) = 342 - 102
18x + 36y - 18x - 6y = 240
30y = 240
To find y, we divide both sides of the equation by 30:
y = 240 / 30
y = 8
Now that we know y, we can substitute it back into the first equation to find x:
x + 2 * 8 = 19
x + 16 = 19
x = 3
Therefore, a small candle holder holds 3 candles and a large candle holder holds 8 candles.