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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineThe 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 1717 regular wreaths and 1111 deluxe wreaths, which used a total of 100100 pinecones. After lunch, they assembled 1717 regular wreaths and 2020 deluxe wreaths, using a total of 154154 pinecones. How many pinecones are they putting on each wreath?\newlineThe regular wreaths each have ___\_\_\_ pinecones on them, and the large ones each have ___\_\_\_ pinecones.

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineThe 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 1717 regular wreaths and 1111 deluxe wreaths, which used a total of 100100 pinecones. After lunch, they assembled 1717 regular wreaths and 2020 deluxe wreaths, using a total of 154154 pinecones. How many pinecones are they putting on each wreath?\newlineThe regular wreaths each have ___\_\_\_ pinecones on them, and the large ones each have ___\_\_\_ pinecones.
  1. Define variables: Define variables for the number of pinecones on each type of wreath. Let rr be the number of pinecones on each regular wreath, and dd be the number of pinecones on each deluxe wreath.
  2. Write equations: Write the system of equations based on the given information.\newlineBefore lunch, the club made 1717 regular wreaths and 1111 deluxe wreaths using 100100 pinecones:\newline17r+11d=10017r + 11d = 100\newlineAfter lunch, they made 1717 regular wreaths and 2020 deluxe wreaths using 154154 pinecones:\newline17r+20d=15417r + 20d = 154
  3. Solve using elimination: Solve the system of equations using the substitution or elimination method.\newlineWe can use the elimination method by multiplying the first equation by −1-1 to align the coefficients of rr:\newline−17r−11d=−100-17r - 11d = -100\newline17r+20d=15417r + 20d = 154
  4. Eliminate rr: Add the two equations to eliminate rr.\(\newline\)(−17r−11d-17r - 11d) + (17r+20d17r + 20d) = −100+154-100 + 154\(\newline\)0r+9d=540r + 9d = 54\(\newline\)9d=549d = 54
  5. Solve for d: Solve for d.\newlined=549d = \frac{54}{9}\newlined=6d = 6
  6. Substitute and solve for rr: Substitute the value of dd back into one of the original equations to solve for rr.17r+11(6)=10017r + 11(6) = 10017r+66=10017r + 66 = 10017r=100−6617r = 100 - 6617r=3417r = 34
  7. Final solution: Solve for rr.r=3417r = \frac{34}{17}r=2r = 2

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