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Q3(iii):
Check which of the following are solutions of the equation $x - 2y = 4$ and which are not:
(iii) $(4, 0)$
Solution :
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.
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(iv): Check which of the following are solutions of the equation $x - 2y = 4$ and which are not: (iv) $(\sqrt{2}, 4\sqrt{2})$
- Q3(v): Check which of the following are solutions of the equation $x - 2y = 4$ and which are not: (v) $(1, 1)$
- 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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