In a shop the cost of 2 pencils and 3 erasers is ₹9 and the cost of 4 pencils and 6 erasers is ₹18. Find the cost of each pencil and each eraser. [Equations: $2x + 3y = 9$ (1) and $4x + 6y = 18$ (2).]

Step 1: Define variables

Let $x$ be the cost of one pencil and $y$ be the cost of one eraser. This helps us set up our equations based on the given information.

Step 2: Formulate equations

According to the problem statement, the cost of 2 pencils and 3 erasers is ₹9, which gives us the first equation. Similarly, the cost of 4 pencils and 6 erasers is ₹18, leading to the second equation.

Step 3: Simplify the second equation

We can simplify the second equation by dividing all terms by 2. This reveals a relationship between the two equations that will help us solve the system. When divided by 2, $4x/2 = 2x$, $6y/2 = 3y$, and $18/2 = 9$, resulting in $2x + 3y = 9$.

Step 4: Compare equations

Upon comparing Equation (1) and Equation (3), we notice that they are identical. This means we essentially have only one unique linear equation for two variables, not a system of two independent equations.

Step 5: Express one variable in terms of the other

Since both equations are the same, we can express one variable in terms of the other. From $2x + 3y = 9$, we can isolate $x$ by first subtracting $3y$ from both sides, giving $2x = 9 - 3y$, and then dividing by 2 to get $x = \frac{9 - 3y}{2}$. This form allows us to find corresponding values of $x$ for any chosen value of $y$.

Step 6: Conclusion about the solution

Since the two given equations are dependent (one is a multiple of the other), they represent the same line. Therefore, any pair of values $(x, y)$ that satisfies one equation will also satisfy the other, leading to infinitely many possible solutions for the cost of each pencil and eraser. For example, if $y = 1$, then $x = \frac{9 - 3(1)}{2} = \frac{6}{2} = 3$. If $y = 3$, then $x = \frac{9 - 3(3)}{2} = \frac{0}{2} = 0$.

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