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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineBen and his cousin Rachel are picking apples in their grandparents' orchard. Ben has filled 88 baskets with apples and is filling them at a rate of 55 baskets per hour. Rachel has 1111 full baskets and will continue picking at 22 baskets per hour. Once the cousins get to the point where they have filled the same number of baskets, they will carry them to the barn and then eat lunch. How long will that take? How much fruit will they have picked by then?\newlineIn __\_\_ hours, the cousins will each have filled __\_\_ baskets with apples.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineBen and his cousin Rachel are picking apples in their grandparents' orchard. Ben has filled 88 baskets with apples and is filling them at a rate of 55 baskets per hour. Rachel has 1111 full baskets and will continue picking at 22 baskets per hour. Once the cousins get to the point where they have filled the same number of baskets, they will carry them to the barn and then eat lunch. How long will that take? How much fruit will they have picked by then?\newlineIn __\_\_ hours, the cousins will each have filled __\_\_ baskets with apples.
  1. Set up equations: Let's set up the equations for Ben and Rachel. Ben starts with 88 baskets and picks 55 more per hour. Rachel starts with 1111 baskets and picks 22 more per hour. We need to find when they have the same number of baskets.\newlineBen's baskets: B=8+5tB = 8 + 5t\newlineRachel's baskets: R=11+2tR = 11 + 2t
  2. Solve for t: Now, we solve for tt when B=RB = R. Setting the equations equal to each other:\newline8+5t=11+2t8 + 5t = 11 + 2t
  3. Subtract and simplify: Subtract 2t2t from both sides to get:\newline8+3t=118 + 3t = 11
  4. Divide by 33: Subtract 88 from both sides:\newline3t=33t = 3
  5. Plug in tt: Divide both sides by 33:t=1t = 1
  6. Check consistency: Plug t=1t = 1 back into either original equation to find the number of baskets each has at that time. Using Ben's equation:\newlineB=8+5(1)=13B = 8 + 5(1) = 13 baskets
  7. Check consistency: Plug t=1t = 1 back into either original equation to find the number of baskets each has at that time. Using Ben's equation:\newlineB=8+5(1)=13B = 8 + 5(1) = 13 baskets Check Rachel's equation for consistency:\newlineR=11+2(1)=13R = 11 + 2(1) = 13 baskets

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