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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineShannon, an office manager, needs to find a courier to deliver a package. The first courier she is considering charges a fee of $15\$15 plus $1\$1 per pound. The second charges $6\$6 plus $4\$4 per pound. Shannon determines that, given her package's weight, the two courier services are equivalent in terms of cost. How much will it cost? What is the weight?\newlineThe two couriers both cost $\$_____ at a package weight of _____ pounds.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineShannon, an office manager, needs to find a courier to deliver a package. The first courier she is considering charges a fee of $15\$15 plus $1\$1 per pound. The second charges $6\$6 plus $4\$4 per pound. Shannon determines that, given her package's weight, the two courier services are equivalent in terms of cost. How much will it cost? What is the weight?\newlineThe two couriers both cost $\$_____ at a package weight of _____ pounds.
  1. Define Variables: Let xx represent the weight of the package in pounds and yy represent the cost to deliver the package.\newlineFor the first courier:\newlineCost = $15\$15 + $1\$1 per pound\newlineThe equation based on the given information is:\newliney=1x+15y = 1x + 15
  2. First Courier: For the second courier:\newlineCost = $6\$6 + $4\$4 per pound\newlineThe equation based on the given information is:\newliney = 44x + 66
  3. Second Courier: System of equations:\newliney=1x+15y = 1x + 15\newliney=4x+6y = 4x + 6\newlineTo find the weight xx at which the cost yy is the same for both couriers, we set the two equations equal to each other:\newline1x+15=4x+61x + 15 = 4x + 6
  4. System of Equations: Solve for xx by isolating the variable.\newlineSubtract 1x1x from both sides:\newline1x+15−1x=4x+6−1x1x + 15 - 1x = 4x + 6 - 1x\newline15=3x+615 = 3x + 6\newlineNow, subtract 66 from both sides:\newline15−6=3x+6−615 - 6 = 3x + 6 - 6\newline9=3x9 = 3x
  5. Solve for x: Divide both sides by 33 to solve for xx:93=3x3\frac{9}{3} = \frac{3x}{3}3=x3 = xSo, the weight of the package is 33 pounds.
  6. Substitute x: Substitute x=3x = 3 into one of the original equations to find the cost yy.\newlineUsing the first courier's equation:\newliney=1x+15y = 1x + 15\newliney=1(3)+15y = 1(3) + 15\newliney=3+15y = 3 + 15\newliney=18y = 18\newlineSo, the cost to deliver the package is $18\$18.

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