Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ An employee at an organic food store is assembling gift baskets for a display. Using wicker baskets, the employee assembled $9$ small baskets and $5$ large baskets, using a total of $180$ pieces of fruit. Using wire baskets, the employee assembled $2$ small baskets and $8$ large baskets, using a total of $164$ pieces of fruit. Assuming that each small basket includes the same amount of fruit, as does every large basket, how many pieces are in each? $\newline$ The small baskets each include $\_$ pieces and the large ones each include $\_$ pieces.

Full solution

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

\newline

An employee at an organic food store is assembling gift baskets for a display. Using wicker baskets, the employee assembled

9

small baskets and

5

large baskets, using a total of

180

pieces of fruit. Using wire baskets, the employee assembled

2

small baskets and

8

large baskets, using a total of

164

pieces of fruit. Assuming that each small basket includes the same amount of fruit, as does every large basket, how many pieces are in each?

\newline

The small baskets each include

\_

pieces and the large ones each include

\_

pieces.

Form Equations: Let's denote the number of pieces of fruit in each small basket as $s$ and in each large basket as $l$ . We can write two equations based on the information given: $\newline$ For wicker baskets: $\newline$ $9s + 5l = 180$ ( $1$ ) $\newline$ For wire baskets: $\newline$ $2s + 8l = 164$ ( $2$ ) $\newline$ We will use these two equations to form a system of equations that we can solve using the elimination method.
Elimination Method: To use elimination, we need to make the coefficients of one of the variables the same in both equations. Let's multiply equation ( $1$ ) by $2$ to match the coefficient of $s$ in equation ( $2$ ). $\newline$ Multiplying equation ( $1$ ) by $2$ , we get: $\newline$ $18s + 10l = 360$ $(3)$ $\newline$ Now we have: $\newline$ $2s + 8l = 164$ $(2)$ $\newline$ $18s + 10l = 360$ $(3)$
Multiply and Subtract: Next, we will subtract equation ( $2$ ) from equation ( $3$ ) to eliminate $s$ and solve for $l$ . $\newline$ $(18s + 10l) - (2s + 8l) = 360 - 164$ $\newline$ $16s + 2l = 196$ $\newline$ This simplifies to: $\newline$ $2l = 196 - 16s$ $\newline$ Since we don't have a value for $s$ yet, we made a mistake. We should have eliminated $s$ completely by subtracting the entire equation ( $2$ ) from equation ( $3$ ).