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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineEmma likes to knit hats and mittens for friends and family. Last fall, she knitted 11 hat and 33 pairs of mittens, which took a total of 4646 hours. This fall, she knitted 11 hat and 11 pair of mittens, which took a total of 1818 hours. If each hat takes the same amount of time and each pair of mittens takes the same time, how long does it take Emma to knit each item?\newlineIt takes Emma _\_ hours to knit a hat and _\_ hours to knit a pair of mittens.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineEmma likes to knit hats and mittens for friends and family. Last fall, she knitted 11 hat and 33 pairs of mittens, which took a total of 4646 hours. This fall, she knitted 11 hat and 11 pair of mittens, which took a total of 1818 hours. If each hat takes the same amount of time and each pair of mittens takes the same time, how long does it take Emma to knit each item?\newlineIt takes Emma _\_ hours to knit a hat and _\_ hours to knit a pair of mittens.
  1. Define Variables: Let's denote the time it takes Emma to knit a hat as hh hours and the time it takes to knit a pair of mittens as mm hours. From the first scenario, we have the equation h+3m=46h + 3m = 46 because she knitted 11 hat and 33 pairs of mittens in 4646 hours.
  2. Form Equations: From the second scenario, we have the equation h+m=18h + m = 18 because she knitted 11 hat and 11 pair of mittens in 1818 hours.
  3. Elimination Method: We now have a system of two equations. We will use elimination to solve for hh and mm. To eliminate hh, we can subtract the second equation from the first equation.
  4. Subtract Equations: Subtracting the second equation from the first, we get (h+3m)−(h+m)=46−18(h + 3m) - (h + m) = 46 - 18, which simplifies to 2m=282m = 28.
  5. Solve for m: Solving for m, we divide both sides of the equation by 22, giving us m=14m = 14. This means it takes Emma 1414 hours to knit a pair of mittens.
  6. Substitute into Equation: Now that we know mm, we can substitute it back into the second equation to find hh. Substituting m=14m = 14 into h+m=18h + m = 18 gives us h+14=18h + 14 = 18.
  7. Final Solution: Solving for hh, we subtract 1414 from both sides of the equation, giving us h=18−14h = 18 - 14, which simplifies to h=4h = 4. This means it takes Emma 44 hours to knit a hat.

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