The glee club needs to raise money for the spring trip to Europe, so the members are assembling holiday wreaths to sell. Before lunch, they assembled 12 regular wreaths and 18 deluxe wreaths, which used a total of 174 pinecones. After lunch, they assembled 17 regular wreaths and 13 deluxe wreaths, using a total of 159 pinecones. How many pinecones are they putting on each wreath?
How many pinecones are on a regular wreath and how many pinecones are on the deluxe wreaths?
Let's assume there are x pinecones on a regular wreath.
So, there would be 12 regular wreaths * x pinecones on each wreath = 12x pinecones for the regular wreaths before lunch.
Similarly, there would be 18 deluxe wreaths * y pinecones on each wreath = 18y pinecones for the deluxe wreaths before lunch.
And the sum of pinecones for both types of wreaths is 12x + 18y = 174. (1)
After lunch, there would be 17 regular wreaths * x pinecones on each wreath = 17x pinecones for the regular wreaths.
Similarly, there would be 13 deluxe wreaths * y pinecones on each wreath = 13y pinecones for the deluxe wreaths.
And the sum of pinecones for both types of wreaths is 17x + 13y = 159. (2)
To solve these equations, let's multiply equation (2) by 2:
34x + 26y = 318. (3)
Now let's multiply equation (1) by 17 and equation (3) by 12, and subtract them:
12(12x + 18y) - 17(34x + 26y) = 12*174 - 17*159
144x + 216y - 578x - 442y = 2088 - 2703
-434x - 226y = -615. (4)
Finally, let's multiply equation (1) by 26, equation (2) by 12, and add them:
26(12x + 18y) + 12(17x + 13y) = 26*174 + 12*159
312x + 468y + 204x + 156y = 4524 + 1908
516x + 624y = 6432. (5)
Now let's solve equations (4) and (5) to find the values of x and y.
Multiplying equation (4) by (-3) and adding it to equation (5):
-3*(-434x - 226y) + 516x + 624y = -3*(-615) + 6432
1302x + 678y + 516x + 624y = 1845 + 6432
1818x + 1302y = 8277. (6)
Multiplying equation (5) by (-2) and adding it to equation (6):
-2*(516x + 624y) + 1818x + 1302y = -2*6432 + 8277
-1032x - 1248y + 1818x + 1302y = -12864 + 8277
786x + 54y = -4587. (7)
Multiplying equation (7) by 37/9, we get:
37/9*(786x + 54y) = 37/9*(-4587)
37*(786/9)x + 37*(54/9)y = -37*4587/9
2938x + 198y = -16877.2222 (8)
Multiplying equation (6) by -2/9, we get:
-2/9*(1818x + 1302y) = -2/9*(-8277)
-2*(1818/9)x - 2*(1302/9)y = 1834
-404x - 260y = 1834 (9)
Multiplying equation (8) by 404 and equation (9) by 2938 and adding them:
404*(2938x + 198y) + 2938*(-404x - 260y) = 404*(-16877.2222) + 2938*1834
1188392x + 80092y - 1188392x - 761680y = -6812182.8568 + 5388372
-681588y = -1423810.8568
Dividing both sides of the equation by -681588 gives us:
y = 2.0894.
Substituting this value of y into equation (4):
-434x - 226(2.0894) = -615
-434x - 474.9784 = -615
-434x = -615 + 474.9784
-434x = -140.0216
x = 0.3223.
Therefore, there are approximately 0.3223 pinecones on a regular wreath.
And there are approximately 2.0894 pinecones on a deluxe wreath.