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Q2(vii):
Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case:
(vii) $y - 2 = 0$
Solution :
Initial Setup & Theoretical Foundation
We are given the linear equation:
$y - 2 = 0$
The objective is to express this equation in the standard general form of a linear equation in two variables, which is defined as:
$ax + by + c = 0$
[Per the fundamental theorem of linear algebra in two dimensions, $x$ and $y$ are the variables, while $a$, $b$, and $c$ are real numbers known as coefficients and constants, respectively. For the equation to represent a line in a two-dimensional Cartesian plane, $a$ and $b$ cannot both be simultaneously zero ($a^2 + b^2 \neq 0$)].
Step 1: Algebraic Manipulation to Standard Form
Observing the given equation $y - 2 = 0$, we note that the variable $x$ is absent. To map this to the two-variable standard form $ax + by + c = 0$, we must introduce the variable $x$ without altering the mathematical truth or value of the original equation.
[By the Zero Property of Multiplication, any real number multiplied by zero equals zero. Therefore, we can introduce the term $0 \cdot x$].
Rewriting the equation with the $x$ term:
$0 \cdot x + y - 2 = 0$
To make the coefficients explicitly clear, we can write the implied coefficient of $y$ (which is $1$) and express the subtraction of $2$ as the addition of a negative constant:
$0x + 1y + (-2) = 0$
Step 2: Extraction of Coefficients
Now, we perform a direct term-by-term comparison between our expanded equation and the standard form:
- Standard Form: $ax + by + c = 0$
- Expanded Equation: $0x + 1y + (-2) = 0$
By equating the corresponding coefficients, we derive:
- The coefficient of $x$ is $a \implies a = 0$
- The coefficient of $y$ is $b \implies b = 1$
- The constant term is $c \implies c = -2$
Geometric Interpretation (Visual Aid)
Geometrically, a linear equation where $a = 0$ and $b \neq 0$ represents a horizontal line parallel to the x-axis. The equation $0x + 1y - 2 = 0$ simplifies to $y = 2$. This means that for any real value of $x$, the value of $y$ remains strictly $2$.
Final Solution: The equation expressed in the standard form $ax + by + c = 0$ is $0x + 1y - 2 = 0$. The corresponding values of the coefficients are $a = 0$, $b = 1$, and $c = -2$.
More Questions from Class 9 Mathematics Linear Equations in Two Variables EXERCISE 4.1
- Q1: The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement. (Take the cost of a notebook to be $x$ and that of a pen to be $y$).
- Q2(i): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (i) $2x + 3y = 9.3\overline{5}$
- Q2(ii): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (ii) $x - \frac{y}{5} - 10 = 0$
- Q2(iii): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (iii) $-2x + 3y = 6$
- Q2(iv): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (iv) $x = 3y$
- Q2(v): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (v) $2x = -5y$
- Q2(vi): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (vi) $3x + 2 = 0$
- Q2(viii): Express the following linear equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$ and $c$ in each case: (viii) $5 = 2x$
CBSE Solutions for Class 9 Mathematics Linear Equations in Two Variables
Chapters in CBSE - Class 9 Mathematics
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