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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineAn employee at an organic food store is assembling gift baskets for a display. Using wicker baskets, the employee assembled 22 small baskets and 1010 large baskets, using a total of 194194 pieces of fruit. Using wire baskets, the employee assembled 99 small baskets and 1010 large baskets, using a total of 243243 pieces of fruit. Assuming that each small basket includes the same amount of fruit, as does every large basket, how many pieces are in each?\newlineThe small baskets each include _\_ pieces and the large ones each include _\_ pieces.

Full solution

Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineAn employee at an organic food store is assembling gift baskets for a display. Using wicker baskets, the employee assembled 22 small baskets and 1010 large baskets, using a total of 194194 pieces of fruit. Using wire baskets, the employee assembled 99 small baskets and 1010 large baskets, using a total of 243243 pieces of fruit. Assuming that each small basket includes the same amount of fruit, as does every large basket, how many pieces are in each?\newlineThe small baskets each include _\_ pieces and the large ones each include _\_ pieces.
  1. Define Variables: Let xx be the number of pieces of fruit in each small basket and yy be the number of pieces of fruit in each large basket.\newlineFirst equation based on wicker baskets: 2x+10y=1942x + 10y = 194
  2. Form Equations: Second equation based on wire baskets: 9x+10y=2439x + 10y = 243
  3. Eliminate Variable: System of equations:\newline2x+10y=1942x + 10y = 194\newline9x+10y=2439x + 10y = 243\newlineWe will eliminate yy by subtracting the first equation from the second.
  4. Solve for x: Subtract the first equation from the second to solve for x.\newline(9x+10y)(2x+10y)=243194(9x + 10y) - (2x + 10y) = 243 - 194\newline9x+10y2x10y=2431949x + 10y - 2x - 10y = 243 - 194\newline7x=497x = 49\newlinex=497x = \frac{49}{7}\newlinex=7x = 7
  5. Solve for y: Substitute x=7x = 7 into the first equation to solve for y.2(7)+10y=1942(7) + 10y = 19414+10y=19414 + 10y = 19410y=1941410y = 194 - 1410y=18010y = 180y=18010y = \frac{180}{10}y=18y = 18

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