Q2(iv):

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

Initial Setup & Objective

We are given the following linear equation in two variables:

$x = 3y$

The 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 ($a^2 + b^2 \neq 0$). After transforming the equation, we must identify the specific numerical values for the coefficients $a$, $b$, and the constant term $c$.

Step 1: Algebraic Manipulation to Standard Form

To convert the given equation into the standard form, all non-zero terms must be transposed to one side of the equality, leaving a zero on the opposite side.

Given:

$x = 3y$

We subtract $3y$ from both sides of the equation [Per the Subtraction Property of Equality, which states that if $A = B$, then $A - C = B - C$]:

$x - 3y = 3y - 3y$

$x - 3y = 0$

Step 2: Explicit Representation of Coefficients

The standard form requires explicit representation of the $x$-term, the $y$-term, and the constant term $c$. We can rewrite the expression $x - 3y = 0$ by explicitly stating the hidden multiplicative identity (1) and the additive identity (0) [By the definition of the standard form of a linear equation]:

$1 \cdot x + (-3) \cdot y + 0 = 0$

Step 3: Identification of Parameters

By directly comparing our expanded equation to the standard form $ax + by + c = 0$, we can map the corresponding coefficients:

• The coefficient of $x$ is $a$. Therefore, $a = 1$.
• The coefficient of $y$ is $b$. Therefore, $b = -3$.
• The constant term is $c$. Since there is no constant term present in the original equation, $c = 0$.

Note: An alternative, equally valid standard form can be achieved by subtracting $x$ from both sides (yielding $-x + 3y + 0 = 0$), which would result in $a = -1$, $b = 3$, and $c = 0$. However, it is conventional mathematical practice to maintain a positive leading coefficient for $x$.

Graphical Representation & Analysis

Because the constant term $c = 0$, the linear equation $x = 3y$ (or $y = \frac{1}{3}x$) represents a straight line that passes directly through the origin $(0,0)$. For every 3 units moved along the x-axis, the line moves 1 unit along the y-axis.

Final Solution: The linear equation expressed in the standard form $ax + by + c = 0$ is $1x - 3y + 0 = 0$. The corresponding values for the coefficients are $a = 1$, $b = -3$, and $c = 0$.