Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Emma likes to knit hats and mittens for friends and family. Last fall, she knitted $1$ hat and $3$ pairs of mittens, which took a total of $46$ hours. This fall, she knitted $1$ hat and $1$ pair of mittens, which took a total of $18$ 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? $\newline$ It takes Emma $\_$ hours to knit a hat and $\_$ hours to knit a pair of mittens.

Full solution

Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

Emma likes to knit hats and mittens for friends and family. Last fall, she knitted

1

hat and

3

pairs of mittens, which took a total of

46

hours. This fall, she knitted

1

hat and

1

pair of mittens, which took a total of

18

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?

\newline

It takes Emma

\_

hours to knit a hat and

\_

hours to knit a pair of mittens.

Define Variables: Let's denote the time it takes Emma to knit a hat as $h$ hours and the time it takes to knit a pair of mittens as $m$ hours. From the first scenario, we have the equation $h + 3m = 46$ because she knitted $1$ hat and $3$ pairs of mittens in $46$ hours.
Form Equations: From the second scenario, we have the equation $h + m = 18$ because she knitted $1$ hat and $1$ pair of mittens in $18$ hours.
Elimination Method: We now have a system of two equations. We will use elimination to solve for $h$ and $m$ . To eliminate $h$ , we can subtract the second equation from the first equation.
Subtract Equations: Subtracting the second equation from the first, we get $(h + 3m) - (h + m) = 46 - 18$ , which simplifies to $2m = 28$ .
Solve for m: Solving for m, we divide both sides of the equation by $2$ , giving us $m = 14$ . This means it takes Emma $14$ hours to knit a pair of mittens.
Substitute into Equation: Now that we know $m$ , we can substitute it back into the second equation to find $h$ . Substituting $m = 14$ into $h + m = 18$ gives us $h + 14 = 18$ .
Final Solution: Solving for $h$ , we subtract $14$ from both sides of the equation, giving us $h = 18 - 14$ , which simplifies to $h = 4$ . This means it takes Emma $4$ hours to knit a hat.