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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineMadelyn is going to ship some gifts to family members, and she is considering two shipping companies. The first shipping company charges a fee of $12\$12 to ship a medium box, plus an additional $3\$3 per pound. A second shipping company charges $15\$15 for the same size of box, plus an additional $2\$2 per pound. At a certain weight, the two shipping methods will cost the same amount. What is that weight? How much will it cost?\newlineAt a weight of ‾\underline{\hspace{2em}} pounds, the two shipping methods both cost $‾\$\underline{\hspace{2em}}.

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineMadelyn is going to ship some gifts to family members, and she is considering two shipping companies. The first shipping company charges a fee of $12\$12 to ship a medium box, plus an additional $3\$3 per pound. A second shipping company charges $15\$15 for the same size of box, plus an additional $2\$2 per pound. At a certain weight, the two shipping methods will cost the same amount. What is that weight? How much will it cost?\newlineAt a weight of ‾\underline{\hspace{2em}} pounds, the two shipping methods both cost $‾\$\underline{\hspace{2em}}.
  1. Write Equation First Shipping Company: Write the equation for the first shipping company.\newlineThe first shipping company charges a $12\$12 fee plus $3\$3 per pound. If we let ww represent the weight in pounds, the cost CC for the first company can be represented by the equation:\newlineC=12+3wC = 12 + 3w
  2. Write Equation Second Shipping Company: Write the equation for the second shipping company.\newlineThe second shipping company charges a $15\$15 fee plus $2\$2 per pound. Using the same variable ww for the weight in pounds, the cost CC for the second company can be represented by the equation:\newlineC=15+2wC = 15 + 2w
  3. Set Equations Equal: Set the two equations equal to each other to find the weight at which the costs are the same. 12+3w=15+2w12 + 3w = 15 + 2w
  4. Solve for w: Solve for w by subtracting 2w2w from both sides of the equation.\newline12+3w−2w=15+2w−2w12 + 3w - 2w = 15 + 2w - 2w\newline12+w=1512 + w = 15
  5. Subtract to Isolate ww: Subtract 1212 from both sides to isolate ww.12+w−12=15−1212 + w - 12 = 15 - 12w=3w = 3
  6. Determine Cost at 33 Pounds: Determine the cost when the weight is 33 pounds using either of the original equations.\newlineUsing the first company's equation:\newlineC=12+3(3)C = 12 + 3(3)\newlineC=12+9C = 12 + 9\newlineC=21C = 21
  7. Verify Cost at 33 Pounds: Verify the cost is the same for the second company when the weight is 33 pounds.\newlineUsing the second company's equation:\newlineC=15+2(3)C = 15 + 2(3)\newlineC=15+6C = 15 + 6\newlineC=21C = 21

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