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Q3(i):
Check which of the following are solutions of the equation $x - 2y = 4$ and which are not:
(i) $(0, 2)$
Solution :
Step 1: Initial Setup & Given Variables
We are given the linear equation in two variables:
$x - 2y = 4$
We need to determine whether the coordinate point $(0, 2)$ is a valid solution to this equation. In the Cartesian coordinate system, an ordered pair $(x, y)$ represents a solution to an equation if and only if substituting the $x$ and $y$ values into the equation results in a true mathematical statement [Per the Fundamental Principle of Algebraic Substitution].
- Given $x$-coordinate (abscissa): $x = 0$
- Given $y$-coordinate (ordinate): $y = 2$
Step 2: Substitution into the Left-Hand Side (LHS)
We isolate the Left-Hand Side (LHS) of the given equation and substitute the given coordinates:
$\text{LHS} = x - 2y$
Substituting $x = 0$ and $y = 2$:
$\text{LHS} = (0) - 2(2)$
$\text{LHS} = 0 - 4$
$\text{LHS} = -4$
Step 3: Comparison with the Right-Hand Side (RHS)
The Right-Hand Side (RHS) of the original equation is a constant:
$\text{RHS} = 4$
Comparing the evaluated LHS with the RHS:
$-4 \neq 4$
$\text{LHS} \neq \text{RHS}$
Because the substitution yields an inequality, the ordered pair $(0, 2)$ does not satisfy the equation.
Step 4: Geometric Interpretation & Verification
Geometrically, a linear equation in two variables represents a straight line on a Cartesian plane. Every point that lies exactly on this line is a solution to the equation. Since $(0, 2)$ does not satisfy the algebraic equation, the point $(0, 2)$ will not lie on the line $x - 2y = 4$.
To graph the line $x - 2y = 4$, we find its intercepts:
- x-intercept (set $y = 0$): $x - 2(0) = 4 \implies x = 4$. Point: $(4, 0)$
- y-intercept (set $x = 0$): $0 - 2y = 4 \implies y = -2$. Point: $(0, -2)$
The visual representation below proves that the point $(0, 2)$ is spatially disconnected from the solution set (the line).
Final Solution: Since substituting $x = 0$ and $y = 2$ results in $\text{LHS} \neq \text{RHS}$ ($-4 \neq 4$), the point $(0, 2)$ 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(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) $(\sqrt{2}, 4\sqrt{2})$
- 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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