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) $5 = 2x$

Given Equation & Initial Setup

We are given the following linear equation in one variable:

$5 = 2x$

Our objective is to express this equation in the general standard form of a linear equation in two variables, which is given by:

$ax + by + c = 0$

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

Step 1: Algebraic Manipulation to Standard Form

To convert the given equation into the standard form, we must transpose all terms to one side of the equality so that the other side becomes zero. [Per the properties of equality, subtracting the same value from both sides preserves the truth value of the equation].

Subtracting $5$ from both sides of the equation:

$5 - 5 = 2x - 5$

$0 = 2x - 5$

By the symmetric property of equality ($A = B \implies B = A$), we can rewrite this as:

$2x - 5 = 0$

Note: Alternatively, subtracting $2x$ from both sides yields $-2x + 5 = 0$. Both forms are mathematically valid, but maintaining a positive leading coefficient is standard convention.

Step 2: Introducing the Second Variable

The standard form requires two variables, $x$ and $y$. Since the variable $y$ is absent from our current equation, it implies that the coefficient of $y$ is exactly zero. [Any real number multiplied by zero is zero, hence $0 \cdot y = 0$].

We explicitly introduce the $y$ term to match the $ax + by + c = 0$ structure:

$2x + 0y - 5 = 0$

Step 3: Identification of Coefficients

Now, we align our derived equation with the general standard form to extract the values of the constants $a$, $b$, and $c$:

• Standard Form: $ax + by + c = 0$
• Derived Equation: $2x + 0y + (-5) = 0$

By direct comparison of the corresponding terms:

• The coefficient of $x$ is $a \implies a = 2$
• The coefficient of $y$ is $b \implies b = 0$
• The constant term is $c \implies c = -5$

Geometric Representation (Graphical Analysis)

A linear equation in two variables where one coefficient is zero represents a line parallel to one of the coordinate axes. The equation $2x - 5 = 0$ simplifies to $x = 2.5$. This represents a vertical line parallel to the Y-axis, intersecting the X-axis at exactly $(2.5, 0)$.

Final Solution: The equation expressed in the standard form $ax + by + c = 0$ is $2x + 0y - 5 = 0$. The corresponding values of the coefficients are $a = 2$, $b = 0$, and $c = -5$.

(Note: The alternative valid form $-2x + 0y + 5 = 0$ yields $a = -2$, $b = 0$, $c = 5$.)