Q3(iii):

Check which of the following are solutions of the equation $x - 2y = 4$ and which are not: (iii) $(4, 0)$

Given Variables & Initial Setup

We are given a linear equation in two variables and an ordered pair representing a point on the Cartesian plane. Our objective is to determine mathematically whether the given point lies on the line defined by the equation.

• The Linear Equation: $x - 2y = 4$
• The Ordered Pair (Point): $(4, 0)$

Step 1: Coordinate Identification & Substitution

An ordered pair is written in the format $(x, y)$. [Per the fundamental principles of coordinate geometry, a point is a solution to an equation if and only if substituting its coordinates into the equation results in a true mathematical statement].

From the given point $(4, 0)$, we extract the specific coordinate values:

• Abscissa ($x$-coordinate): $x = 4$
• Ordinate ($y$-coordinate): $y = 0$

We substitute these values into the Left-Hand Side (LHS) of the given equation.

$\text{LHS} = x - 2y$

$\text{LHS} = (4) - 2(0)$

Step 2: Algebraic Evaluation

Applying the standard order of operations (PEMDAS/BODMAS), we first perform the multiplication, followed by the subtraction.

$\text{LHS} = 4 - 0$

$\text{LHS} = 4$

Step 3: Verification of Equality

We now compare the evaluated Left-Hand Side (LHS) with the Right-Hand Side (RHS) of the original equation.

• $\text{LHS} = 4$
• $\text{RHS} = 4$

Since $\text{LHS} = \text{RHS}$, the equation holds true. [By the Axiom of Equality, the substitution satisfies the linear equation, proving that the point geometrically lies exactly on the line].

Graphical Representation & Verification

To rigorously verify this result, we can map the equation $x - 2y = 4$ and the point $(4, 0)$ onto a Cartesian coordinate system. The line intersects the x-axis precisely at the point $(4, 0)$.

Final Solution: The ordered pair $(4, 0)$ satisfies the equation $x - 2y = 4$. Therefore, $(4, 0)$ is a valid solution to the given linear equation.