Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Isabelle and her friend Mabel are knitting scarves for the homeless. Isabelle had already completed $2$ scarves and has committed to making $7$ more scarves per day. Mabel, who hasn't completed any scarves yet, has more free time and can make $9$ scarves per day. At some point, Mabel 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.

View step-by-step help

Home

Math Problems

Algebra 1

Solve a system of equations using substitution: word problems

Full solution

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

\newline

Isabelle and her friend Mabel are knitting scarves for the homeless. Isabelle had already completed

2

scarves and has committed to making

7

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

9

scarves per day. At some point, Mabel 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 and Equations: Define the variables and write the equations for Isabelle and Mabel. $\newline$ Let $x$ represent the number of days after they start knitting. $\newline$ Isabelle's equation: She starts with $2$ scarves and knits $7$ more per day. $\newline$ Total scarves by Isabelle = Initial scarves + (Scarves per day $*$ Days) $\newline$ $I(x) = 2 + 7x$ $\newline$ Mabel's equation: She starts with $0$ scarves and knits $9$ per day. $\newline$ Total scarves by Mabel = Initial scarves + (Scarves per day $*$ Days) $\newline$ $M(x) = 0 + 9x$
Set Equations Equal: Set the equations equal to each other to find the day when they have the same number of scarves. $\newline$ $I(x) = M(x)$ $\newline$ $2 + 7x = 9x$
Solve for x: Solve for x by isolating the variable. $\newline$ $2 + 7x = 9x$ $\newline$ $7x - 9x = -2$ $\newline$ $-2x = -2$ $\newline$ $x = 1$
Substitute x Value: Substitute the value of $x$ back into either equation to find the total number of scarves each woman has finished. $\newline$ Using Isabelle's equation: $\newline$ $I(1) = 2 + 7(1)$ $\newline$ $I(1) = 2 + 7$ $\newline$ $I(1) = 9$
Verify Mabel's Equation: Verify the result by substituting $x$ into Mabel's equation. $\newline$ $M(1) = 9(1)$ $\newline$ $M(1) = 9$
Answer Prompt: Answer the question prompt with the values found. $\newline$ After $1$ day, each woman will have finished $9$ scarves.