Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ A clothing store is donating socks to various charities. The store gave $5$ regular packs and $5$ value packs to a homeless shelter, which contained a total of $420$ pairs of socks. For foster children, the store donated $5$ regular packs and $6$ value packs, which equaled $456$ pairs. How many pairs of socks are in each pack? $\newline$ There are $\_$ pairs of socks in each regular pack and $\_$ pairs in each value pack.

Full solution

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

\newline

A clothing store is donating socks to various charities. The store gave

5

regular packs and

5

value packs to a homeless shelter, which contained a total of

420

pairs of socks. For foster children, the store donated

5

regular packs and

6

value packs, which equaled

456

pairs. How many pairs of socks are in each pack?

\newline

There are

\_

pairs of socks in each regular pack and

\_

pairs in each value pack.

Define Variables: Let's define the variables: $\newline$ Let $r$ be the number of pairs of socks in a regular pack. $\newline$ Let $v$ be the number of pairs of socks in a value pack. $\newline$ We can write two equations based on the information given: $\newline$ For the homeless shelter: $5r + 5v = 420$ $\newline$ For the foster children: $5r + 6v = 456$
Write Equations: Now, we will use the elimination method to solve the system of equations. We can eliminate one of the variables by subtracting one equation from the other. $\newline$ Subtract the first equation from the second equation: $\newline$ $(5r + 6v) - (5r + 5v) = 456 - 420$ $\newline$ This simplifies to: $\newline$ $v = 36$
Elimination Method: Now that we have the value of $v$ , we can substitute it back into one of the original equations to find the value of $r$ . Let's substitute $v$ into the first equation: $5r + 5(36) = 420$ $5r + 180 = 420$
Substitute Value: Next, we will solve for $r$ : $\newline$ $5r = 420 - 180$ $\newline$ $5r = 240$ $\newline$ $r = \frac{240}{5}$ $\newline$ $r = 48$
Solve for r: We have found the values for both $r$ and $v$ : $\newline$ $r = 48$ pairs of socks in each regular pack $\newline$ $v = 36$ pairs of socks in each value pack