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Q3(v):
Check which of the following are solutions of the equation $x - 2y = 4$ and which are not:
(v) $(1, 1)$
Solution :
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 $x - 2y = 4$.
More Questions from Class 9 Mathematics Linear Equations in Two Variables EXERCISE 4.2
- Q1: Which one of the following options is true, and why? $y = 3x + 5$ has
- Q2(i): Write four solutions for each of the following equations: (i) $2x + y = 7$
- Q2(ii): Write four solutions for each of the following equations: (ii) $\pi x + y = 9$
- Q2(iii): Write four solutions for each of the following equations: (iii) $x = 4y$
- Q3(i): Check which of the following are solutions of the equation $x - 2y = 4$ and which are not: (i) $(0, 2)$
- Q3(ii): Check which of the following are solutions of the equation $x - 2y = 4$ and which are not: (ii) $(2, 0)$
- Q3(iii): Check which of the following are solutions of the equation $x - 2y = 4$ and which are not: (iii) $(4, 0)$
- Q3(iv): Check which of the following are solutions of the equation $x - 2y = 4$ and which are not: (iv) $(\sqrt{2}, 4\sqrt{2})$
- Q4: Find the value of $k$, if $x = 2$, $y = 1$ is a solution of the equation $2x + 3y = k$.
CBSE Solutions for Class 9 Mathematics Linear Equations in Two Variables
Chapters in CBSE - Class 9 Mathematics
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