Q.The total cost of 11 pencils and 3 erasers is $\text{\text{₹}}\,64$ and the total cost of 8 pencils and 3 erasers is $\text{\text{₹}}\,49$. Find the cost of each pencil and each eraser by Cramer's Rule.

Set up $11x+3y=64$, $8x+3y=49$; by Cramer's rule $x=45/9=5$ and $y=27/9=3$, i.e. pencil ₹5, eraser ₹3.

Step 1 — Form the equations. Let pencil $=x$, eraser $=y$:

$11x+3y=64,\qquad 8x+3y=49.$

Step 2 — Coefficient determinant:

$\Delta=\begin{vmatrix}11 & 3\\ 8 & 3\end{vmatrix}=(11)(3)-(3)(8)=33-24=9\ ( eq0),$

so a unique solution exists.

Step 3 — Replace columns:

$\Delta_x=\begin{vmatrix}64 & 3\\ 49 & 3\end{vmatrix}=(64)(3)-(3)(49)=192-147=45,$

$\Delta_y=\begin{vmatrix}11 & 64\\ 8 & 49\end{vmatrix}=(11)(49)-(64)(8)=539-512=27.$

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