Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Amy and her sister Katie are making baby blankets to sell at a boutique. Amy has already completed $12$ blankets and can finish $3$ more blankets per day. Katie has already completed $7$ blankets and can finish $8$ more blankets per day. At some point, they will have completed the same number of blankets. How many blankets will each woman have made? How long will that take? $\newline$ Each woman will have finished blankets in days. $\newline$ Submit

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Solve a system of equations using substitution: word problems

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

\newline

Amy and her sister Katie are making baby blankets to sell at a boutique. Amy has already completed

12

blankets and can finish

3

more blankets per day. Katie has already completed

7

blankets and can finish

8

more blankets per day. At some point, they will have completed the same number of blankets. How many blankets will each woman have made? How long will that take?

\newline

Each woman will have finished blankets in days.

\newline

Submit

Define Variables: Let's define the variables. $\newline$ Let $x$ represent the number of days that have passed. $\newline$ Let $y$ represent the total number of blankets made by each woman. $\newline$ Amy's rate of making blankets is $3$ per day, and she starts with $12$ . $\newline$ Katie's rate of making blankets is $8$ per day, and she starts with $7$ .
Write Equations: Write the equations based on the given information. $\newline$ For Amy: $y = 3x + 12$ (since she makes $3$ blankets per day and has already made $12$ ) $\newline$ For Katie: $y = 8x + 7$ (since she makes $8$ blankets per day and has already made $7$ )
Substitution to Solve: Use substitution to solve for $x$ . Since both $y$ 's represent the same number of blankets, we can set the equations equal to each other: $3x + 12 = 8x + 7$
Solve for x: Solve for x. $\newline$ Subtract $3x$ from both sides: $\newline$ $3x + 12 - 3x = 8x + 7 - 3x$ $\newline$ $12 = 5x + 7$ $\newline$ Now, subtract $7$ from both sides: $\newline$ $12 - 7 = 5x + 7 - 7$ $\newline$ $5 = 5x$ $\newline$ Divide both sides by $5$ : $\newline$ $\frac{5}{5} = \frac{5x}{5}$ $\newline$ $x = 1$
Find Total Blankets: Find the total number of blankets made by each woman after $x$ days. $\newline$ Since $x = 1$ , we substitute $x$ back into either of the original equations to find $y$ . $\newline$ Using Amy's equation: $y = 3x + 12$ $\newline$ $y = 3(1) + 12$ $\newline$ $y = 3 + 12$ $\newline$ $y = 15$
Verify Solution: Verify the result with Katie's equation. $\newline$ Using Katie's equation: $y = 8x + 7$ $\newline$ $y = 8(1) + 7$ $\newline$ $y = 8 + 7$ $\newline$ $y = 15$ $\newline$ Since the result is the same, our solution is consistent.