Question

9. The cost of 4 dozen pencil, 3 dozen pens, 2 dozen erasers
is 60. The cost of 2 dozen pencils, 4 dozen pen and 6 dozen
eraser is Rs. 90 where as the cost of 6 dozen pencil,
2 dozen pen and 3 dozen eraser is Rs. 70. Find the cost
of each item per dozen by using matices.​

Answer 1

The cost of pencils, pens & erasers per dozen by using matrices is Rs. 5, Rs. 8 & Rs. 8 respectively.

Step-by-step explanation:

Let the cost of pencils, pens &  erasers per dozen be denoted as Rs. “x”, Rs. “y” & Rs. “z” respectively.

According to the question let’s write the equations first:

4x + 3y + 2z = 60 ….. (i)

2x + 4y + 6z = 90 ….. (ii)

6x + 2y + 3z = 70 …… (iii)

Since we are asked to solve the cost of each item by using matrices, so let’s arrange the equation in matrix form:

$\left[\begin{array}{ccc}4&3&2\\2&4&6\\6&2&3\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}60\\90\\70\end{array}\right]$  i.e., AX = B

R₂ → [R₂ - 1/2R₁] & R₃ → [R₃ - 3/2R₁]

$\left[\begin{array}{ccc}4&3&2\\0&5/2&5\\0&-5/2&0\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}60\\60\\-20\end{array}\right]$

R₃→ [R₃ + R₂]

$\left[\begin{array}{ccc}4&3&2\\0&5/2&5\\0&0&5\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}60\\60\\40\end{array}\right]$

From the above matrix that we got, we can now write,  

5z = 40

⇒ z = 40/5 = 8 …… (iv)

And,

5/2y + 5z = 60

⇒5/2y + (5*8) = 60

⇒ 5/2y = 60 - 40

⇒ y = 20*2/5 = 8 ….. (v)

Now, by substituting the values of y & z from (iv) & (v) in eq. (i), we get

4x + (3*8) + (2*8) = 60

⇒ 4x = 60 – 16 – 24

⇒ x = 20/4 = 5

Thus, the cost of 1 dozen pencil is Rs. 5, 1 dozen pens is Rs. 8 and 1 dozen eraser is Rs. 8.

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Answer 2

the cost of 36 dozen pencil 3 such pencil