Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Baldwin and Gavin, both teachers, are adding books to their class libraries. Baldwin's classroom started out with a collection of only $7$ books, but he plans to purchase an additional $2$ books per week. Gavin's library started out with $9$ books, and he has enough money in his budget to purchase another $1$ book per week. At some point, the two teachers' libraries will contain the same number of books. How many weeks will that take? How many books will each class have? $\newline$ After $\_$ weeks, the two teachers' libraries will each have $\_$ books.

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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

Baldwin and Gavin, both teachers, are adding books to their class libraries. Baldwin's classroom started out with a collection of only

7

books, but he plans to purchase an additional

2

books per week. Gavin's library started out with

9

books, and he has enough money in his budget to purchase another

1

book per week. At some point, the two teachers' libraries will contain the same number of books. How many weeks will that take? How many books will each class have?

\newline

After

\_

weeks, the two teachers' libraries will each have

\_

books.

Define Variables: Let's define the variables: $\newline$ Let $B$ represent the total number of books in Baldwin's library. $\newline$ Let $G$ represent the total number of books in Gavin's library. $\newline$ Let $w$ represent the number of weeks that have passed. $\newline$ We can write two equations to represent the situation: $\newline$ For Baldwin: $B = 7 + 2w$ (since he starts with $7$ books and adds $2$ each week) $\newline$ For Gavin: $G = 9 + w$ (since he starts with $9$ books and adds $1$ each week) $\newline$ We want to find the value of $w$ when $B$ equals $G$ .
Set Up Equations: Now we set up the system of equations: $\newline$ $1$ ) $B = 7 + 2w$ $\newline$ $2$ ) $G = 9 + w$ $\newline$ Since we are looking for when $B$ equals $G$ , we can set the two equations equal to each other: $\newline$ $7 + 2w = 9 + w$
Solve for w: Next, we solve for w using substitution or elimination. In this case, we can simply subtract $w$ from both sides to isolate $w$ : $7 + 2w - w = 9 + w - w$ This simplifies to: $7 + w = 9$
Determine Books After $2$ Weeks: Now we solve for $w$ : $\newline$ $7 + w = 9$ $\newline$ Subtract $7$ from both sides: $\newline$ $w = 9 - 7$ $\newline$ $w = 2$
Check Solution: We have found that after $2$ weeks, the two teachers' libraries will have the same number of books. Now we need to determine how many books each class will have. $\newline$ We can substitute $w$ back into either original equation. Let's use Baldwin's equation: $\newline$ $B = 7 + 2w$ $\newline$ $B = 7 + 2(2)$ $\newline$ $B = 7 + 4$ $\newline$ $B = 11$
Check Solution: We have found that after $2$ weeks, the two teachers' libraries will have the same number of books. Now we need to determine how many books each class will have. $\newline$ We can substitute $w$ back into either original equation. Let's use Baldwin's equation: $\newline$ $B = 7 + 2w$ $\newline$ $B = 7 + 2(2)$ $\newline$ $B = 7 + 4$ $\newline$ $B = 11$ To check our work, we should also substitute $w$ into Gavin's equation to ensure it gives us the same number of books: $\newline$ $G = 9 + w$ $\newline$ $G = 9 + 2$ $\newline$ $G = $11$
Check Solution: We have found that after $2$ weeks, the two teachers' libraries will have the same number of books. Now we need to determine how many books each class will have.$\newline$We can substitute $w$ back into either original equation. Let's use Baldwin's equation:$\newline$$B = 7 + 2w$$\newline$$B = 7 + 2(2)$$\newline$$B = 7 + 4$$\newline$$B = 11$To check our work, we should also substitute $w$ into Gavin's equation to ensure it gives us the same number of books:$\newline$$G = 9 + w$$\newline$$G = 9 + 2$$\newline$$G = 11$Both Baldwin and Gavin will have $w$$0$ books in their libraries after $2$ weeks. This confirms our solution is correct.