Answer: Option 'd' is correct.
Step-by-step explanation:
Let the number of small hat be 'x'
Let the number of medium hat be 'y'.
Let the number of large hat be 'x'.
Cost of each small hat = $5.50
Cost of each medium hat = $ 6.00
Cost of each large hat = $7.00
According to this, our equations would be
[tex]x+y+z=47---(1)\\\\5.5x+6y+7z=\$302------(2)[/tex]
And we have given that He purchases three times as many medium hats as small hats.
So, our equation would be
[tex]y=3x\\\\-3x+y=0-------(3)[/tex]
so, the matrix would be
Ax=b
[tex]\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][/tex]
So, According to this, we can rewrite it as :
[tex]x=A^{-1}B[/tex]
Here, Inverse of A would be
[tex]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][/tex]
So, the value of 'X' would be written as
[tex]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][/tex]
So, we using calculator, we get that
x = 6
y = 18
z = 23
So, Number of large hats purchased by the coach are 23.
Hence, Option 'd' is correct.
