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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineA local service organization is wrapping gifts at the mall to raise money for charity. Yesterday, they wrapped 3333 small gifts and 2323 large gifts, earning a total of $240\$240. Today, they wrapped 1919 small gifts and 3535 large gifts, and earned $334\$334. How much did they charge to wrap the gifts?\newlineThe organization charges $\$_____ to wrap a small gift and $\$_____ to wrap a large one.

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineA local service organization is wrapping gifts at the mall to raise money for charity. Yesterday, they wrapped 3333 small gifts and 2323 large gifts, earning a total of $240\$240. Today, they wrapped 1919 small gifts and 3535 large gifts, and earned $334\$334. How much did they charge to wrap the gifts?\newlineThe organization charges $\$_____ to wrap a small gift and $\$_____ to wrap a large one.
  1. Define Charges: Let's denote the charge for wrapping a small gift as xx dollars and for a large gift as yy dollars.\newlineWe can write two equations based on the given information.\newlineYesterday's earnings: 3333 small gifts and 2323 large gifts earned $240\$240.\newlineToday's earnings: 1919 small gifts and 3535 large gifts earned $334\$334.
  2. Translate into Equations: Translate the given information into a system of equations.\newlineFor yesterday's earnings: 33x+23y=24033x + 23y = 240\newlineFor today's earnings: 19x+35y=33419x + 35y = 334
  3. Solve Using Elimination: Solve the system of equations using any method. We will use the elimination method.\newlineFirst, we need to make the coefficients of either xx or yy the same in both equations. Let's multiply the first equation by 1919 and the second equation by 3333 to eliminate xx.\newline19(33x+23y)=19(240)19(33x + 23y) = 19(240)\newline33(19x+35y)=33(334)33(19x + 35y) = 33(334)
  4. Perform Multiplication: Perform the multiplication from Step 33.\newline627x+437y=4560627x + 437y = 4560\newline627x+1155y=11022627x + 1155y = 11022
  5. Subtract Equations: Subtract the second equation from the first to eliminate xx.\newline(627x+437y)(627x+1155y)=456011022(627x + 437y) - (627x + 1155y) = 4560 - 11022\newline627x+437y627x1155y=5462627x + 437y - 627x - 1155y = -5462\newline718y=5462-718y = -5462
  6. Solve for y: Solve for y.\newline718y=5462-718y = -5462\newliney=5462718y = \frac{-5462}{-718}\newliney=7.61y = 7.61\newlineSince the charge for wrapping a gift is unlikely to be a non-integer value, we should check our calculations for any errors.

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