Q2(iii):

Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (iii) $-2x + 3y = 6$

Given Variables & Initial Setup

We are given the following linear equation in two variables:

$-2x + 3y = 6$

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

$ax + by + c = 0$

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

Step 1: Algebraic Manipulation to Standard Form

To convert the given equation into the standard form, we must ensure that all terms (variables and constants) are collected on the left-hand side (LHS) of the equation, leaving exactly zero on the right-hand side (RHS).

• Original Equation: $-2x + 3y = 6$
• Transformation: Apply the Subtraction Property of Equality [Subtracting $6$ from both sides of the equation to nullify the RHS].
• $-2x + 3y - 6 = 6 - 6$
• $-2x + 3y - 6 = 0$

Step 2: Identification of Coefficients

Now, we align our transformed equation with the standard theoretical model to extract the corresponding coefficients.

Standard Form: $ax + by + c = 0$

Derived Equation: $(-2)x + (3)y + (-6) = 0$

By direct comparison of the corresponding terms, we identify the values:

• The coefficient of $x$, denoted as $a$, is $-2$.
• The coefficient of $y$, denoted as $b$, is $3$.
• The constant term, denoted as $c$, is $-6$.

Note on Mathematical Equivalence: Multiplying the entire equation by $-1$ yields $2x - 3y + 6 = 0$. In this equivalent standard form, the coefficients would be $a = 2$, $b = -3$, and $c = 6$. Both representations are mathematically rigorous and correct, though the direct transposition (yielding $a = -2$) is the primary expected procedure.

Graphical Representation of the Linear Equation

To provide a complete analytical perspective, the equation $-2x + 3y = 6$ represents a straight line on a Cartesian plane. The intercepts are calculated as follows:

• x-intercept: Set $y = 0 \implies -2x = 6 \implies x = -3$. Coordinate: $(-3, 0)$
• y-intercept: Set $x = 0 \implies 3y = 6 \implies y = 2$. Coordinate: $(0, 2)$

Final Solution: The linear equation expressed in the standard form $ax + by + c = 0$ is $-2x + 3y - 6 = 0$. The corresponding values of the coefficients are $a = -2$, $b = 3$, and $c = -6$.