Q1:

The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement. (Take the cost of a notebook to be $x$ and that of a pen to be $y$).

Step 1: Variable Assignment & Initial Setup

To construct a mathematical model of the given real-world scenario, we first define the unknown quantities using algebraic variables. The problem explicitly provides the variable assignments:

• Let the cost of a single notebook be represented by the variable $x$.
• Let the cost of a single pen be represented by the variable $y$.

[These variables represent continuous, non-negative rational numbers, as they denote monetary cost: $x, y \ge 0$].

Step 2: Mathematical Translation of the Statement

The core logical condition provided is: "The cost of a notebook is twice the cost of a pen."

We break down this statement into its algebraic equivalents:

• "The cost of a notebook" translates to $x$.
• "is" translates to the equality operator $=$.
• "twice the cost of a pen" translates to the scalar multiplication $2 \times y$, or simply $2y$.

Step 3: Formulation of the Linear Equation

Equating the translated expressions yields the direct algebraic relationship:

$x = 2y$

[This equation demonstrates a direct proportional relationship between the cost of the notebook and the cost of the pen, where the constant of proportionality is $2$].

Step 4: Standardization of the Equation

The standard form of a linear equation in two variables is expressed as:

$ax + by + c = 0$

where $a$, $b$, and $c$ are real numbers, and $a$ and $b$ are not both zero ($a^2 + b^2 \neq 0$).

To convert our formulated equation $x = 2y$ into the standard form, we apply the Subtraction Property of Equality by subtracting $2y$ from both sides of the equation:

$x - 2y = 2y - 2y$

$x - 2y = 0$

Here, the coefficients are $a = 1$, $b = -2$, and the constant term is $c = 0$.

Graphical Representation (Visual Verification)

The linear equation $x - 2y = 0$ can be graphed on a Cartesian coordinate system. Because the variables represent cost, the relevant domain and range are restricted to the first quadrant ($x \ge 0, y \ge 0$). The line passes through the origin $(0,0)$, confirming that if a pen costs $0$, a notebook also costs $0$.

Final Solution: The linear equation in two variables representing the given statement is $x - 2y = 0$.