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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineCarmen likes to make desserts for bake sales. Last month, she made 33 batches of brownies and 33 batches of cookies, which called for 1818 eggs total. The month before, she baked 33 batches of brownies and 11 batch of cookies, which required a total of 1010 eggs. How many eggs did Carmen use for a batch of each dessert?\newlineCarmen uses _\_ eggs to make a batch of brownies and _\_ eggs to make a batch of cookies.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineCarmen likes to make desserts for bake sales. Last month, she made 33 batches of brownies and 33 batches of cookies, which called for 1818 eggs total. The month before, she baked 33 batches of brownies and 11 batch of cookies, which required a total of 1010 eggs. How many eggs did Carmen use for a batch of each dessert?\newlineCarmen uses _\_ eggs to make a batch of brownies and _\_ eggs to make a batch of cookies.
  1. Define Variables: Let's denote the number of eggs used for a batch of brownies as bb and the number of eggs used for a batch of cookies as cc. From the first situation, we have the equation 3b+3c=183b + 3c = 18.
  2. Form Equations: From the second situation, we have the equation 3b+1c=103b + 1c = 10.
  3. Elimination Method: We now have a system of two equations:\newline11. 3b+3c=183b + 3c = 18\newline22. 3b+1c=103b + 1c = 10\newlineWe will use elimination to solve this system. To eliminate bb, we can subtract the second equation from the first.
  4. Solve for c: Subtracting the second equation from the first gives us 3b+3c−(3b+1c)=18−103b + 3c - (3b + 1c) = 18 - 10, which simplifies to 2c=82c = 8.
  5. Substitute cc into Equation: Dividing both sides of 2c=82c = 8 by 22 gives us c=4c = 4. This means Carmen uses 44 eggs to make a batch of cookies.
  6. Solve for bb: Now we substitute c=4c = 4 into the second equation 3b+1c=103b + 1c = 10 to find bb. This gives us 3b+1(4)=103b + 1(4) = 10, which simplifies to 3b+4=103b + 4 = 10.
  7. Solve for b: Now we substitute c=4c = 4 into the second equation 3b+1c=103b + 1c = 10 to find bb. This gives us 3b+1(4)=103b + 1(4) = 10, which simplifies to 3b+4=103b + 4 = 10. Subtracting 44 from both sides of 3b+4=103b + 4 = 10 gives us 3b=63b = 6.
  8. Solve for b: Now we substitute c=4c = 4 into the second equation 3b+1c=103b + 1c = 10 to find b. This gives us 3b+1(4)=103b + 1(4) = 10, which simplifies to 3b+4=103b + 4 = 10. Subtracting 44 from both sides of 3b+4=103b + 4 = 10 gives us 3b=63b = 6. Dividing both sides of 3b=63b = 6 by 33 gives us b=2b = 2. This means Carmen uses 3b+1c=103b + 1c = 1000 eggs to make a batch of brownies.

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