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) $y - 2 = 0$

Initial Setup & Theoretical Foundation

We are given the linear equation:

$y - 2 = 0$

The objective is to express this equation in the standard general form of a linear equation in two variables, which is defined as:

$ax + by + c = 0$

[Per the fundamental theorem of linear algebra in two dimensions, $x$ and $y$ are the variables, while $a$, $b$, and $c$ are real numbers known as coefficients and constants, respectively. For the equation to represent a line in a two-dimensional Cartesian plane, $a$ and $b$ cannot both be simultaneously zero ($a^2 + b^2 \neq 0$)].

Step 1: Algebraic Manipulation to Standard Form

Observing the given equation $y - 2 = 0$, we note that the variable $x$ is absent. To map this to the two-variable standard form $ax + by + c = 0$, we must introduce the variable $x$ without altering the mathematical truth or value of the original equation.

[By the Zero Property of Multiplication, any real number multiplied by zero equals zero. Therefore, we can introduce the term $0 \cdot x$].

Rewriting the equation with the $x$ term:

$0 \cdot x + y - 2 = 0$

To make the coefficients explicitly clear, we can write the implied coefficient of $y$ (which is $1$) and express the subtraction of $2$ as the addition of a negative constant:

$0x + 1y + (-2) = 0$

Step 2: Extraction of Coefficients

Now, we perform a direct term-by-term comparison between our expanded equation and the standard form:

• Standard Form: $ax + by + c = 0$
• Expanded Equation: $0x + 1y + (-2) = 0$

By equating the corresponding coefficients, we derive:

• The coefficient of $x$ is $a \implies a = 0$
• The coefficient of $y$ is $b \implies b = 1$
• The constant term is $c \implies c = -2$

Geometric Interpretation (Visual Aid)

Geometrically, a linear equation where $a = 0$ and $b \neq 0$ represents a horizontal line parallel to the x-axis. The equation $0x + 1y - 2 = 0$ simplifies to $y = 2$. This means that for any real value of $x$, the value of $y$ remains strictly $2$.

Final Solution: The equation expressed in the standard form $ax + by + c = 0$ is $0x + 1y - 2 = 0$. The corresponding values of the coefficients are $a = 0$, $b = 1$, and $c = -2$.