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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 $y$).
Solution :
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 $x - 2y = 0$.
More Questions from Class 9 Mathematics Linear Equations in Two Variables EXERCISE 4.1
- Q2(i): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (i) $2x + 3y = 9.3\overline{5}$
- Q2(ii): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (ii) $x - \frac{y}{5} - 10 = 0$
- Q2(iii): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (iii) $-2x + 3y = 6$
- Q2(iv): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (iv) $x = 3y$
- Q2(v): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (v) $2x = -5y$
- Q2(vi): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (vi) $3x + 2 = 0$
- Q2(vii): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (vii) $y - 2 = 0$
- Q2(viii): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (viii) $5 = 2x$
CBSE Solutions for Class 9 Mathematics Linear Equations in Two Variables
Chapters in CBSE - Class 9 Mathematics
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