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) $2x + 3y = 9.3\overline{5}$

Initial Setup & Theoretical Foundation

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

$ax + by + c = 0$

where $a$, $b$, and $c$ are real numbers ($a, b, c \in \mathbb{R}$), and the coefficients $a$ and $b$ are not simultaneously zero (often denoted as $a^2 + b^2 \neq 0$).

We are given the following linear equation:

$2x + 3y = 9.3\overline{5}$

Step 1: Transposition to Standard Form

To map the given equation to the standard form, we must collect all terms on the left-hand side of the equality, leaving exactly zero on the right-hand side.

[Per the Subtraction Property of Equality], we subtract the constant term $9.3\overline{5}$ from both sides of the equation. Note that $9.3\overline{5}$ is a rational real number (a non-terminating, repeating decimal), and it is treated algebraically as a single constant value.

$2x + 3y - 9.3\overline{5} = 9.3\overline{5} - 9.3\overline{5}$

$2x + 3y - 9.3\overline{5} = 0$

Step 2: Coefficient Extraction and Comparative Analysis

To strictly match the $ax + by + c = 0$ structure, we can conceptually express the subtraction of the constant as the addition of a negative constant:

$2x + 3y + (-9.3\overline{5}) = 0$

By applying the principle of polynomial identity, we equate the corresponding coefficients from our derived equation directly to the standard form variables:

• The coefficient of the $x$-term corresponds to $a \implies a = 2$
• The coefficient of the $y$-term corresponds to $b \implies b = 3$
• The constant term corresponds to $c \implies c = -9.3\overline{5}$

Final Solution: The linear equation expressed in standard form is $2x + 3y - 9.3\overline{5} = 0$. The corresponding coefficient values are $a = 2$, $b = 3$, and $c = -9.3\overline{5}$.