Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. An employee at a party store is assembling balloon bouquets. F... Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. An employee at a party store is assembling balloon bouquets. For a graduation party, he assembled 6 small balloon bouquets and 10 large balloon bouquets, which used a total of 252 balloons. Then, for a Father's Day celebration, he used 196 balloons to assemble 1 small balloon bouquet and 9 large balloon bouquets. How many balloons are in each bouquet? The small balloon bouquet uses ___ balloons and the large one uses ___ balloons. Read more

Steps to Solve

1. Define Variables

Let ( x ) be the number of balloons in a small bouquet, and ( y ) be the number of balloons in a large bouquet.

2. Set Up the First Equation

Using the information from the graduation party:

[ 6x + 10y = 252 ]

3. Set Up the Second Equation

Using the information from the Father's Day celebration:

[ 1x + 9y = 196 ]

4. Form the Augmented Matrix

The augmented matrix from the equations ( 6x + 10y = 252 ) and ( 1x + 9y = 196 ) is

$$ \begin{bmatrix} 6 & 10 & | & 252 \ 1 & 9 & | & 196 \end{bmatrix} $$

5. Apply Row Operations

Perform row operations to convert the augmented matrix to reduced row echelon form. Start with:

• Row 1: ( R_1 ) stays the same.
• Row 2: ( R_2 = R_2 - \frac{1}{6} R_1 ).

The new matrix is:

$$ \begin{bmatrix} 6 & 10 & | & 252 \ 0 & \frac{43}{6} & | & 115 \end{bmatrix} $$

6. Continue Simplifying

Now simplify further. Multiply ( R_2 ) by ( \frac{6}{43} ):

$$ \begin{bmatrix} 6 & 10 & | & 252 \ 0 & 1 & | & \frac{690}{43} \end{bmatrix} $$

Then perform row reduction on ( R_1 ):

$$ R_1 = R_1 - 10R_2 $$

The final matrix is:

$$ \begin{bmatrix} 1 & 0 & | & ... \ 0 & 1 & | & ... \end{bmatrix} $$

7. Back Substitute to Find Values

Using the final augmented matrix, back substitute to find ( x ) and ( y ).