Linear Equations in two Variables — NCERT Solutions

Given: Cost of a notebook = ₹x, Cost of a pen = ₹y.

Statement: The cost of a notebook is twice the cost of a pen.

Forming the equation:
$$x = 2y$$

This can be written in standard form as:
$$x - 2y = 0$$

or equivalently $1 \cdot x + (-2) \cdot y + 0 = 0$.

Answer: The required linear equation in two variables is $x = 2y$ (or $x - 2y = 0$).

The standard form is $ax + by + c = 0$.

(i) $2x + 3y = 9.35$

Rewriting: $2x + 3y - 9.35 = 0$

Here, $a = 2,\; b = 3,\; c = -9.35$.

(ii) $x - \dfrac{y}{5} - 10 = 0$

This is already in the required form: $1 \cdot x + \left(-\dfrac{1}{5}\right)y + (-10) = 0$

Here, $a = 1,\; b = -\dfrac{1}{5},\; c = -10$.

(iii) $-2x + 3y = 6$

Rewriting: $-2x + 3y - 6 = 0$

Here, $a = -2,\; b = 3,\; c = -6$.

(iv) $x = 3y$

Rewriting: $x - 3y = 0$, i.e., $1 \cdot x + (-3)y + 0 = 0$

Here, $a = 1,\; b = -3,\; c = 0$.

(v) $2x = -5y$

Rewriting: $2x + 5y = 0$, i.e., $2x + 5y + 0 = 0$

Here, $a = 2,\; b = 5,\; c = 0$.

(vi) $3x + 2 = 0$

Rewriting: $3x + 0 \cdot y + 2 = 0$

Here, $a = 3,\; b = 0,\; c = 2$.

(vii) $y - 2 = 0$

Rewriting: $0 \cdot x + 1 \cdot y + (-2) = 0$

Here, $a = 0,\; b = 1,\; c = -2$.

(viii) $5 = 2x$

Rewriting: $2x = 5 \Rightarrow 2x - 5 = 0$, i.e., $2x + 0 \cdot y + (-5) = 0$

Here, $a = 2,\; b = 0,\; c = -5$.

Correct option: (iii) infinitely many solutions.

Reason: The equation $y = 3x + 5$ is a linear equation in two variables $x$ and $y$. For every real value we assign to $x$, we get a corresponding real value of $y$. Since there are infinitely many real numbers to substitute for $x$, the equation has infinitely many solutions.

For example:

• $x = 0 \Rightarrow y = 5$ → solution $(0, 5)$
• $x = 1 \Rightarrow y = 8$ → solution $(1, 8)$
• $x = -1 \Rightarrow y = 2$ → solution $(-1, 2)$

Thus, a linear equation in two variables always has infinitely many solutions.