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 on a regular wreath and how many pinecones are on a deluxe wreath?

Answer 1

Let r be the number of pinecones on a regular wreath and d be the number of pinecones on a deluxe wreath.
We know that r * 12 + d * 18 = 174 and r * 17 + d * 13 = 159.
We can multiply the second equation by 12 to make the coefficients of r the same in both equations: 204r + 156d = (159 * 12) = 1908.
We can then subtract the first equation from this equation to eliminate r: 204r + 156d - (12r + 18d) = 1908 - 174.
This simplifies to: 192r + 138d = 1734.
We can divide both sides by 6 to simplify the equation further: 32r + 23d = 289.
We can then multiply the first equation by 23 and the second equation by 18 to eliminate d: 23r * 12 + 23d * 18 = 23 * 174 = 4002 and 18r * 17 + 18d * 13 = 18 * 159 = 2862.
This simplifies to: 276r + 414d = 4002 and 306r + 234d = 2862.
We can subtract the second equation from the first equation to eliminate r: 276r + 414d - (306r + 234d) = 4002 - 2862.
This simplifies to: -30r + 180d = 1140.
We can divide both sides by 6 to simplify the equation further: -5r + 30d = 190.
We can multiply the second equation by 5 to eliminate d: 5 * (-5r + 30d) = 5 * 190.
This simplifies to: -25r + 150d = 950.
We can then subtract the first equation from this equation to eliminate d: -25r + 150d - (-30r + 180d) = 950 - 1140.
This simplifies to: 5r - 30d = -190.
We can multiply this equation by 6 to eliminate r: 6 * (5r - 30d) = 6 * -190.
This simplifies to: 30r - 180d = -1140.
We can add this equation to the previous equation to eliminate d: -30r + 180d + 30r - 180d = -190 + (-1140).
This simplifies to: 0 = -1330.
Since we arrived at an inconsistent statement, there is no solution to this system of equations.
Therefore, it is not possible to determine the number of pinecones on a regular wreath and a deluxe wreath.