Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. An employee at an organic food store is assembling gift basket... Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. An employee at an organic food store is assembling gift baskets for a display. Using wicker baskets, the employee assembled 2 small baskets and 4 large baskets, using a total of 76 pieces of fruit. Using wire baskets, the employee assembled 2 small baskets and 8 large baskets, using a total of 132 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? The small baskets each include ______ pieces and the large ones each include ______ pieces. Read more

Steps to Solve

1. Define Variables
   Let ( x ) be the number of pieces of fruit in each small basket and ( y ) be the number of pieces of fruit in each large basket.

2. Formulate the First Equation
   Using wicker baskets, the employee assembled 2 small baskets and 4 large baskets, totaling 76 pieces of fruit. This gives: $$ 2x + 4y = 76 $$

3. Formulate the Second Equation
   Using wire baskets, the employee assembled 2 small baskets and 8 large baskets, totaling 132 pieces of fruit. Thus we have: $$ 2x + 8y = 132 $$

4. Set Up the Augmented Matrix
   The equations can be represented in augmented matrix form: $$ \begin{bmatrix} 2 & 4 & | & 76 \ 2 & 8 & | & 132 \end{bmatrix} $$

5. Row Reduction
   We will reduce the augmented matrix. First, divide the first row by 2: $$ \begin{bmatrix} 1 & 2 & | & 38 \ 2 & 8 & | & 132 \end{bmatrix} $$
   Now subtract ( 2 \times ) the first row from the second row: $$ \begin{bmatrix} 1 & 2 & | & 38 \ 0 & 4 & | & 56 \end{bmatrix} $$

6. Solve for y
   Next, divide the second row by 4: $$ \begin{bmatrix} 1 & 2 & | & 38 \ 0 & 1 & | & 14 \end{bmatrix} $$

7. Back Substitution to Solve for x
   Substituting ( y = 14 ) back into the first row: $$ x + 2(14) = 38 $$
   This simplifies to: $$ x + 28 = 38 $$
   Subtract 28 from both sides to find: $$ x = 10 $$

8. Conclusion
   Thus, the small baskets each include ( 10 ) pieces of fruit, and the large ones each include ( 14 ) pieces.