Q2(v):

Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (v) $2x = -5y$

Initial Setup & Theoretical Foundation

We are given the linear equation:

$2x = -5y$

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 simultaneously (i.e., $a^2 + b^2 \neq 0$).

Step 1: Algebraic Transposition

To convert the given equation into the standard form, all non-zero terms must be collected on the left-hand side (LHS) of the equation, leaving zero on the right-hand side (RHS).

We apply the Addition Property of Equality [which states that adding the same quantity to both sides of an equation preserves the equality]. We add $5y$ to both sides of the equation:

$2x + 5y = -5y + 5y$

$2x + 5y = 0$

Step 2: Structuring the Standard Form

The standard form requires three distinct terms: an $x$-term, a $y$-term, and a constant term $c$. In our current equation, $2x + 5y = 0$, the constant term is absent. Mathematically, an absent term implies a coefficient or value of zero. We explicitly introduce zero as the constant term to perfectly match the $ax + by + c = 0$ template:

$2x + 5y + 0 = 0$

Step 3: Coefficient Extraction

By directly comparing our structured equation with the general form $ax + by + c = 0$, we can extract the corresponding real number coefficients:

• The coefficient of $x$ corresponds to $a$. Thus, $a = 2$.
• The coefficient of $y$ corresponds to $b$. Thus, $b = 5$.
• The constant term corresponds to $c$. Thus, $c = 0$.

Graphical Representation (Geometric Interpretation)

A linear equation in two variables geometrically represents a straight line on a Cartesian plane. Because the constant term $c = 0$, the line must pass through the origin $(0,0)$. We can verify this by finding two points that satisfy $2x + 5y = 0$:

• If $x = 0$, then $5y = 0 \implies y = 0$. Point: $(0, 0)$
• If $x = 5$, then $10 + 5y = 0 \implies 5y = -10 \implies y = -2$. Point: $(5, -2)$
• If $x = -5$, then $-10 + 5y = 0 \implies 5y = 10 \implies y = 2$. Point: $(-5, 2)$

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