Question

three friends went to the store. meg bought 5 apples, 6 bananas, and 2 pears for a total of $16.05. al bought 3 apples, 7 bananas, and 4 pears for a total of $15.12. jan bought 8 apples, 1 banana, and 11 pears for a total of $27.42. complete the matrix equation that could be used to determine the price of one apple, a, one banana, b, and one pear, p. matrix with empty boxes options: 5, 6, 2, 16.05, 3, 7, 4, 8, 1, 0, 11, 15.12, 27.42, a, b, p

Explanation:

Step1: Define Variables and Equations

Let \( a \) = price of one apple, \( b \) = price of one banana, \( p \) = price of one pear.
Meg's purchase: \( 5a + 6b + 2p = 16.05 \)
Al's purchase: \( 3a + 7b + 4p = 15.12 \)
Jan's purchase: \( 8a + 1b + 11p = 27.42 \)

Step2: Form Matrix Equation

A matrix equation for a system \( Ax = B \) has:

• \( A \): Coefficient matrix (rows = purchases, columns = \( a, b, p \))
• \( x \): Variable vector \(

$$\begin{bmatrix} a \\ b \\ p \end{bmatrix}$$

\)

• \( B \): Constant vector (total costs)

Coefficient Matrix \( A \):

• Row 1 (Meg): \( [5, 6, 2] \)
• Row 2 (Al): \( [3, 7, 4] \)
• Row 3 (Jan): \( [8, 1, 11] \)

Variable Vector \( x \):

\(

$$\begin{bmatrix} a \\ b \\ p \end{bmatrix}$$

\)

Constant Vector \( B \):

\(

$$\begin{bmatrix} 16.05 \\ 15.12 \\ 27.42 \end{bmatrix}$$

\)

Answer:

The matrix equation is:

$$
LATEXBLOCK0

LATEXBLOCK1
=
LATEXBLOCK2
$$

Filling the blanks in the given matrix:

• Coefficient matrix (left):

Row 1: \( 5, 6, 2 \)
Row 2: \( 3, 7, 4 \)
Row 3: \( 8, 1, 11 \)

• Variable vector (middle): \(

$$\begin{bmatrix} a \\ b \\ p \end{bmatrix}$$

\)

• Constant vector (right): \(

$$\begin{bmatrix} 16.05 \\ 15.12 \\ 27.42 \end{bmatrix}$$

\)