Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Isabella and her friend Nicole are knitting scarves for the homeless. Isabella had already completed $5$ scarves and has committed to making $8$ more scarves per day. Nicole, who hasn't completed any scarves yet, has more free time and can make $9$ scarves per day. At some point, Nicole will catch up and they will both have completed the same number of scarves. How long will that take? How many scarves will each woman have finished? $\newline$ After $\_$ days, each woman will have finished $\_$ scarves.

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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

Isabella and her friend Nicole are knitting scarves for the homeless. Isabella had already completed

5

scarves and has committed to making

8

more scarves per day. Nicole, who hasn't completed any scarves yet, has more free time and can make

9

scarves per day. At some point, Nicole will catch up and they will both have completed the same number of scarves. How long will that take? How many scarves will each woman have finished?

\newline

After

\_

days, each woman will have finished

\_

scarves.

Define Variables: Let's define the variables: $\newline$ Let $x$ represent the number of days after which they will have completed the same number of scarves. $\newline$ Let $y$ represent the total number of scarves each woman will have finished after $x$ days. $\newline$ Isabella's scarf equation: $\newline$ Isabella starts with $5$ scarves and knits $8$ more per day. $\newline$ $y = 8x + 5$
Isabella's Scarf Equation: Nicole's scarf equation: $\newline$ Nicole starts with $0$ scarves and knits $9$ per day. $\newline$ $y = 9x$
Nicole's Scarf Equation: Now we have two equations: $\newline$ $1$ ) $y = 8x + 5$ (Isabella's equation) $\newline$ $2$ ) $y = 9x$ (Nicole's equation) $\newline$ We can use substitution to find when Nicole will catch up to Isabella by setting the two equations equal to each other since $y$ represents the same total number of scarves for both women. $\newline$ $8x + 5 = 9x$
Substitution: Solve for $x$ : $8x + 5 = 9x$ $8x - 9x = -5$ $-x = -5$ $x = 5$ So, it will take $5$ days for Nicole to catch up to Isabella.
Solve for x: Now we need to find the total number of scarves each woman will have finished after $5$ days. We can substitute $x = 5$ into either of the original equations. Let's use Isabella's equation: $\newline$ $y = 8x + 5$ $\newline$ $y = 8(5) + 5$ $\newline$ $y = 40 + 5$ $\newline$ $y = 45$ $\newline$ After $5$ days, each woman will have finished $45$ scarves.