A coach purchases 47 hats for his players and their families at a total cost of $302. The cost of a small hat is $5.50. A medium

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A coach purchases 47 hats for his players and their families at a total cost of $302. The cost of a small hat is $5.50. A medium

hat costs $6.00. A large hat costs $7.00. He purchases three times as many medium hats as small hats. Using matrices, how many large hats did the coach purchase?
a. 6

b. 11

c. 18

d. 23

Mathematics

2 answers:

Leona [9.2K] · Answer 1 · 2026-03-27T16:55:19+03:00

Answer: Option 'd' is correct.

Step-by-step explanation:

Let 'x' denote the number of small hats,

Let 'y' represent the number of medium hats.

Let 'z' signify the number of large hats.

Each small hat costs $5.50

Each medium hat is priced at $6.00Each large hat costs $7.00

According to this, we can formulate the equations

$x+y+z=47---(1)\\\\5.5x+6y+7z=\$302------(2)$

We know that he bought three times the number of medium hats compared to small hats.

Thus, we establish the equation

$y=3x\\\\-3x+y=0-------(3)$

Consequently, the matrix can be expressed as

Ax=b

$\left[\begin{array}{ccc}1&1&1\\5.5&6&7\\-3&1&0\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] =\left[\begin{array}{ccc}47\\302\\0\end{array}\right]$

Therefore, we can rewrite this as:

$x=A^{-1}B$

The inverse of A would be

$A^{-1}=\left[\begin{array}{ccc}\dfrac{14}{9}&\dfrac{-2}{9}&\dfrac{-2}{9}\\\\\dfrac{14}{3}&\dfrac{-2}{3}&\dfrac{1}{3}\\\\\dfrac{-47}{9}&\dfrac{8}{9}&\dfrac{-1}{9}\end{array}\right]$

Thus, the value of 'X' can be expressed as

$x=\left[\begin{array}{ccc}\dfrac{14}{9}&\dfrac{-2}{9}&\dfrac{-2}{9}\\\\\dfrac{14}{3}&\dfrac{-2}{3}&\dfrac{1}{3}\\\\\dfrac{-47}{9}&\dfrac{8}{9}&\dfrac{-1}{9}\end{array}\right]\left[\begin{array}{ccc}47\\\\302\\\\0\end{array}\right] =\left[\begin{array}{ccc}6\\\\18\\\\23\end{array}\right]$

Using a calculator, we find that

x = 6

y = 18

z = 23

Therefore, the number of large hats acquired by the coach is 23.

Hence, Option 'd' is indeed correct.

tester [8.8K] · Answer 2 · 2026-03-27T16:57:10+03:00

To start, we will assign variable definitions:
x: small hat
y: medium hat
z: large hat
Next, the system of equations can be expressed as:
x + y + z = 47
5.50x + 6y + 7z = 302
-3x + y = 0
We can rewrite the system as a matrix equation:
Ax = b
Where,
A = [1 1 1; 5.50 6 7; -3 1 0]
b = [47; 302; 0]
x = [x; y; z]
By solving the system, we find:
x = 6
y = 18
z = 23
Thus, the coach bought 23 large hats
d. 23