Q3(v):

Check which of the following are solutions of the equation $x - 2y = 4$ and which are not: (v) $(1, 1)$

Given Variables & Initial Setup

We are tasked with determining whether the coordinate point $(1, 1)$ is a valid solution to the given linear equation in two variables. The foundational elements of our analysis are:

• The Linear Equation: $x - 2y = 4$
• The Test Coordinate: $(1, 1)$

Step 1: Deconstructing the Coordinate Pair

In the Cartesian coordinate system, an ordered pair is universally represented in the format $(x, y)$. [By the fundamental axiom of coordinate geometry], the first value corresponds to the abscissa (the $x$-coordinate) and the second value corresponds to the ordinate (the $y$-coordinate).

Extracting the values from our test coordinate $(1, 1)$:

• $x = 1$
• $y = 1$

Step 2: Substitution into the Left-Hand Side (LHS)

To verify if the point lies on the line defined by the equation, we must substitute the extracted $x$ and $y$ values into the Left-Hand Side (LHS) of the equation. [Per the algebraic property of substitution, a point is a solution if and only if it satisfies the equality].

The LHS of the equation is given by:

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

Substituting $x = 1$ and $y = 1$:

$\text{LHS} = (1) - 2(1)$

$\text{LHS} = 1 - 2$

$\text{LHS} = -1$

Step 3: Comparative Analysis of LHS and RHS

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

• $\text{LHS} = -1$
• $\text{RHS} = 4$

Comparing the two values:

$-1 \neq 4$

Since the Left-Hand Side does not equal the Right-Hand Side ($\text{LHS} \neq \text{RHS}$), the coordinate pair $(1, 1)$ fails to satisfy the linear equation.

Step 4: Graphical Verification

To provide rigorous geometric proof, we can visualize the linear equation $x - 2y = 4$ and the point $(1, 1)$ on a Cartesian plane. A point is a solution to an equation if and only if it lies exactly on the line representing that equation.

As demonstrated in the high-precision graph above, the blue line represents the infinite set of solutions for $x - 2y = 4$. The red point representing the coordinate $(1, 1)$ clearly lies outside the trajectory of the line, confirming our algebraic derivation geometrically.

Final Conclusion

Because the substitution of $x = 1$ and $y = 1$ results in a mathematical contradiction ($-1 \neq 4$), the point does not satisfy the equation.

Final Solution: The coordinate $(1, 1)$ is not a solution to the equation $x - 2y = 4$.