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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineTwo workers in a holiday boutique are filling stockings with small gifts and candy. Beth has already filled 44 stockings and will continue to fill them at a rate of 22 stockings per hour. Aaron, who just arrived to help, can fill 44 stockings per hour. At some point, Aaron will catch up with Beth and they will have completed the same number of stockings. How many stockings will each worker have filled by then? How long will that take?\newlineThe workers will each have filled _\_ stockings in _\_ hours.

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineTwo workers in a holiday boutique are filling stockings with small gifts and candy. Beth has already filled 44 stockings and will continue to fill them at a rate of 22 stockings per hour. Aaron, who just arrived to help, can fill 44 stockings per hour. At some point, Aaron will catch up with Beth and they will have completed the same number of stockings. How many stockings will each worker have filled by then? How long will that take?\newlineThe workers will each have filled _\_ stockings in _\_ hours.
  1. Define variables: Let's define the variables. Let xx be the number of hours Aaron works until he catches up with Beth. Since Beth has already filled 44 stockings and continues at a rate of 22 per hour, the total number of stockings she will have filled by the time Aaron catches up is 4+2x4 + 2x. Aaron fills stockings at a rate of 44 per hour, so he will have filled 4x4x stockings in xx hours.
  2. Set up equation: Set up the equation based on the information that Aaron will catch up with Beth. At that point, they will have filled the same number of stockings, so we can equate Beth's total to Aaron's total:\newline4+2x=4x4 + 2x = 4x
  3. Solve for x: Solve the equation for x. Subtract 2x2x from both sides to isolate the variable on one side:\newline4+2x−2x=4x−2x4 + 2x - 2x = 4x - 2x\newline4=2x4 = 2x\newlineDivide both sides by 22 to solve for x:\newline42=2x2\frac{4}{2} = \frac{2x}{2}\newlinex=2x = 2
  4. Calculate total stockings: Now that we know x=2x = 2, we can find out how many stockings each worker will have filled. For Beth:\newlineTotal stockings filled by Beth = 4+2x=4+2(2)=4+4=84 + 2x = 4 + 2(2) = 4 + 4 = 8\newlineFor Aaron:\newlineTotal stockings filled by Aaron = 4x=4(2)=84x = 4(2) = 8
  5. Check solution: Check the solution by substituting xx back into the original equation to ensure both sides are equal:\newline4+2(2)=4(2)4 + 2(2) = 4(2)\newline4+4=84 + 4 = 8\newline8=88 = 8\newlineSince both sides are equal, the solution is correct.

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