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) $3x + 2 = 0$

Initial Setup & Given Equation

We are given the linear equation:

$3x + 2 = 0$

The objective is to express this equation in the standard two-variable form, $ax + by + c = 0$, and to identify the corresponding real coefficients $a$, $b$, and $c$.

Step 1: Algebraic Transformation to Standard Form

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

$ax + by + c = 0$

where $x$ and $y$ are variables, and $a$, $b$, and $c$ are real numbers such that $a$ and $b$ are not simultaneously zero [Per the fundamental definition of a linear equation in two variables].

Observing the given equation $3x + 2 = 0$, we note the absence of the $y$-variable. To represent this equation in a two-variable format without altering its mathematical equivalence, we introduce the variable $y$ with a coefficient of $0$. [Since $0 \cdot y = 0$ for any real number $y$, the additive identity property ensures the equation's value remains unchanged].

Rewriting the equation yields:

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

Step 2: Extraction of Coefficients

By aligning our transformed equation directly with the standard form, we can map the corresponding coefficients:

• Standard Form: $ax + by + c = 0$
• Transformed Equation: $3x + 0y + 2 = 0$

Comparing the terms positionally:

• The coefficient of $x$ is $a = 3$.
• The coefficient of $y$ is $b = 0$.
• The constant term is $c = 2$.

Geometric Interpretation (Graphical Representation)

Geometrically, solving the equation $3x + 2 = 0$ for $x$ yields $x = -\frac{2}{3}$. In a two-dimensional Cartesian coordinate system, an equation of the form $x = k$ (where $k$ is a constant) represents a vertical line parallel to the $y$-axis. The line intersects the $x$-axis at the coordinate $(-\frac{2}{3}, 0)$. Because $y$ can take any real value while $x$ remains constant, the coefficient of $y$ is effectively zero.

Final Conclusion

Final Solution: The equation expressed in the standard form $ax + by + c = 0$ is $3x + 0y + 2 = 0$. The corresponding coefficients are $a = 3$, $b = 0$, and $c = 2$.