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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) $-2x + 3y = 6$
Solution :
Given Variables & Initial Setup
We are given the following linear equation in two variables:
$-2x + 3y = 6$
The objective is to express this equation in the general (standard) form of a linear equation in two variables, which is defined as:
$ax + by + c = 0$
where $a$, $b$, and $c$ are real numbers, and $a$ and $b$ are not both zero (i.e., $a^2 + b^2 \neq 0$).
Step 1: Algebraic Manipulation to Standard Form
To convert the given equation into the standard form, we must ensure that all terms (variables and constants) are collected on the left-hand side (LHS) of the equation, leaving exactly zero on the right-hand side (RHS).
- Original Equation: $-2x + 3y = 6$
- Transformation: Apply the Subtraction Property of Equality [Subtracting $6$ from both sides of the equation to nullify the RHS].
- $-2x + 3y - 6 = 6 - 6$
- $-2x + 3y - 6 = 0$
Step 2: Identification of Coefficients
Now, we align our transformed equation with the standard theoretical model to extract the corresponding coefficients.
Standard Form: $ax + by + c = 0$
Derived Equation: $(-2)x + (3)y + (-6) = 0$
By direct comparison of the corresponding terms, we identify the values:
- The coefficient of $x$, denoted as $a$, is $-2$.
- The coefficient of $y$, denoted as $b$, is $3$.
- The constant term, denoted as $c$, is $-6$.
Note on Mathematical Equivalence: Multiplying the entire equation by $-1$ yields $2x - 3y + 6 = 0$. In this equivalent standard form, the coefficients would be $a = 2$, $b = -3$, and $c = 6$. Both representations are mathematically rigorous and correct, though the direct transposition (yielding $a = -2$) is the primary expected procedure.
Graphical Representation of the Linear Equation
To provide a complete analytical perspective, the equation $-2x + 3y = 6$ represents a straight line on a Cartesian plane. The intercepts are calculated as follows:
- x-intercept: Set $y = 0 \implies -2x = 6 \implies x = -3$. Coordinate: $(-3, 0)$
- y-intercept: Set $x = 0 \implies 3y = 6 \implies y = 2$. Coordinate: $(0, 2)$
Final Solution: The linear equation expressed in the standard form $ax + by + c = 0$ is $-2x + 3y - 6 = 0$. The corresponding values of the coefficients are $a = -2$, $b = 3$, and $c = -6$.
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(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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