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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineAn online boutique is having a special on personalized baby items. On Monday, they sold 1212 personalized baby blankets and 1111 personalized hooded towels, for a total of $569\$569 in receipts. The following day, they received orders for 11 personalized baby blanket and 1111 personalized hooded towels, which brought in a total of $239\$239. How much does each item sell for?\newlineBlankets sell for $\$_____ apiece, and hooded towels sell for $\$_____ apiece.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineAn online boutique is having a special on personalized baby items. On Monday, they sold 1212 personalized baby blankets and 1111 personalized hooded towels, for a total of $569\$569 in receipts. The following day, they received orders for 11 personalized baby blanket and 1111 personalized hooded towels, which brought in a total of $239\$239. How much does each item sell for?\newlineBlankets sell for $\$_____ apiece, and hooded towels sell for $\$_____ apiece.
  1. Define Prices: Let's denote the price of each personalized baby blanket as bb and the price of each personalized hooded towel as tt. From the first day's sales, we have the equation 12b+11t=56912b + 11t = 569. This represents the sale of 1212 blankets and 1111 towels for a total of $569\$569.
  2. First Day Sales: From the second day's sales, we have the equation 1b+11t=2391b + 11t = 239. This represents the sale of 11 blanket and 1111 towels for a total of $239\$239.
  3. Second Day Sales: We now have a system of two equations. To solve using elimination, we need to eliminate one of the variables, bb or tt. We choose to eliminate bb because it has a coefficient of 11 in the second equation, which makes it easier to manipulate.
  4. Eliminate Variable: To eliminate bb, we multiply the second equation by −12-12, the negative of the coefficient of bb in the first equation. This gives us the new equation −12b−132t=−2868-12b - 132t = -2868.
  5. Solve for t: We now add the first equation to the new second equation to eliminate bb. This gives us −121t=−2299-121t = -2299. Solving for tt, we get t=2299121t = \frac{2299}{121}.
  6. Calculate t: After calculating t=2299121t = \frac{2299}{121}, we find that t=19t = 19. This means that each personalized hooded towel sells for $19\$19.
  7. Substitute and Solve for b: We substitute t=19t = 19 into the second original equation 1b+11t=2391b + 11t = 239 and solve for b. This gives us 1b+11(19)=2391b + 11(19) = 239, which simplifies to 1b+209=2391b + 209 = 239.
  8. Substitute and Solve for b: We substitute t=19t = 19 into the second original equation 1b+11t=2391b + 11t = 239 and solve for b. This gives us 1b+11(19)=2391b + 11(19) = 239, which simplifies to 1b+209=2391b + 209 = 239. Subtracting 209209 from both sides of the equation 1b+209=2391b + 209 = 239, we get 1b=301b = 30. Therefore, b=30b = 30. This means that each personalized baby blanket sells for $30\$30.

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