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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineKate, an office manager, needs to find a courier to deliver a package. The first courier she is considering charges a fee of $14\$14 plus $1\$1 per pound. The second charges $13\$13 plus $2\$2 per pound. Kate determines that, given her package's weight, the two courier services are equivalent in terms of cost. What is the weight? How much will it cost?\newlineAt a package weight of __\_\_ pounds, the two couriers both cost $______\$\_\_\_\_\_\_.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineKate, an office manager, needs to find a courier to deliver a package. The first courier she is considering charges a fee of $14\$14 plus $1\$1 per pound. The second charges $13\$13 plus $2\$2 per pound. Kate determines that, given her package's weight, the two courier services are equivalent in terms of cost. What is the weight? How much will it cost?\newlineAt a package weight of __\_\_ pounds, the two couriers both cost $______\$\_\_\_\_\_\_.
  1. Define Variables: Let's define the variables:\newlineLet xx be the weight of the package in pounds.\newlineLet yy be the total cost for the courier service.\newlineThe first courier's cost can be represented by the equation: y=14+1×xy = 14 + 1 \times x.\newlineThe second courier's cost can be represented by the equation: y=13+2×xy = 13 + 2 \times x.\newlineWe need to write a system of equations to represent the situation.
  2. Write Equations: Now we have the system of equations:\newline11) y=14+xy = 14 + x\newline22) y=13+2xy = 13 + 2x\newlineWe will use substitution to solve this system. Since both equations equal yy, we can set them equal to each other to find the value of xx.\newline14+x=13+2x14 + x = 13 + 2x
  3. Use Substitution: Next, we solve for xx:14+x=13+2x14 + x = 13 + 2xSubtract xx from both sides:14=13+x14 = 13 + xSubtract 1313 from both sides:1=x1 = xSo, the weight of the package is 11 pound.
  4. Solve for x: Now that we have the value of x, we can substitute it back into either of the original equations to find the cost y. We'll use the first equation:\newliney = 1414 + x\newliney = 1414 + 11\newliney = 1515\newlineSo, the cost for both couriers is $15\$15.

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