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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) $2x + 3y = 9.3\overline{5}$
Solution :
Initial Setup & Theoretical Foundation
The general standard form of a linear equation in two variables is defined algebraically as:
$ax + by + c = 0$
where $a$, $b$, and $c$ are real numbers ($a, b, c \in \mathbb{R}$), and the coefficients $a$ and $b$ are not simultaneously zero (often denoted as $a^2 + b^2 \neq 0$).
We are given the following linear equation:
$2x + 3y = 9.3\overline{5}$
Step 1: Transposition to Standard Form
To map the given equation to the standard form, we must collect all terms on the left-hand side of the equality, leaving exactly zero on the right-hand side.
[Per the Subtraction Property of Equality], we subtract the constant term $9.3\overline{5}$ from both sides of the equation. Note that $9.3\overline{5}$ is a rational real number (a non-terminating, repeating decimal), and it is treated algebraically as a single constant value.
$2x + 3y - 9.3\overline{5} = 9.3\overline{5} - 9.3\overline{5}$
$2x + 3y - 9.3\overline{5} = 0$
Step 2: Coefficient Extraction and Comparative Analysis
To strictly match the $ax + by + c = 0$ structure, we can conceptually express the subtraction of the constant as the addition of a negative constant:
$2x + 3y + (-9.3\overline{5}) = 0$
By applying the principle of polynomial identity, we equate the corresponding coefficients from our derived equation directly to the standard form variables:
- The coefficient of the $x$-term corresponds to $a \implies a = 2$
- The coefficient of the $y$-term corresponds to $b \implies b = 3$
- The constant term corresponds to $c \implies c = -9.3\overline{5}$
Final Solution: The linear equation expressed in standard form is $2x + 3y - 9.3\overline{5} = 0$. The corresponding coefficient values are $a = 2$, $b = 3$, and $c = -9.3\overline{5}$.
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(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(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) $y - 2 = 0$
- 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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