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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) $3x + 2 = 0$
Solution :
Initial Setup & Given Equation
We are given the linear equation:
$3x + 2 = 0$
The objective is to express this equation in the standard two-variable form, $ax + by + c = 0$, and to identify the corresponding real coefficients $a$, $b$, and $c$.
Step 1: Algebraic Transformation to Standard Form
The standard form of a linear equation in two variables is defined as:
$ax + by + c = 0$
where $x$ and $y$ are variables, and $a$, $b$, and $c$ are real numbers such that $a$ and $b$ are not simultaneously zero [Per the fundamental definition of a linear equation in two variables].
Observing the given equation $3x + 2 = 0$, we note the absence of the $y$-variable. To represent this equation in a two-variable format without altering its mathematical equivalence, we introduce the variable $y$ with a coefficient of $0$. [Since $0 \cdot y = 0$ for any real number $y$, the additive identity property ensures the equation's value remains unchanged].
Rewriting the equation yields:
$3x + 0 \cdot y + 2 = 0$
Step 2: Extraction of Coefficients
By aligning our transformed equation directly with the standard form, we can map the corresponding coefficients:
- Standard Form: $ax + by + c = 0$
- Transformed Equation: $3x + 0y + 2 = 0$
Comparing the terms positionally:
- The coefficient of $x$ is $a = 3$.
- The coefficient of $y$ is $b = 0$.
- The constant term is $c = 2$.
Geometric Interpretation (Graphical Representation)
Geometrically, solving the equation $3x + 2 = 0$ for $x$ yields $x = -\frac{2}{3}$. In a two-dimensional Cartesian coordinate system, an equation of the form $x = k$ (where $k$ is a constant) represents a vertical line parallel to the $y$-axis. The line intersects the $x$-axis at the coordinate $(-\frac{2}{3}, 0)$. Because $y$ can take any real value while $x$ remains constant, the coefficient of $y$ is effectively zero.
Final Conclusion
Final Solution: The equation expressed in the standard form $ax + by + c = 0$ is $3x + 0y + 2 = 0$. The corresponding coefficients are $a = 3$, $b = 0$, 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(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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