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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) $2x = -5y$
Solution :
Initial Setup & Theoretical Foundation
We are given the linear equation:
$2x = -5y$
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 simultaneously (i.e., $a^2 + b^2 \neq 0$).
Step 1: Algebraic Transposition
To convert the given equation into the standard form, all non-zero terms must be collected on the left-hand side (LHS) of the equation, leaving zero on the right-hand side (RHS).
We apply the Addition Property of Equality [which states that adding the same quantity to both sides of an equation preserves the equality]. We add $5y$ to both sides of the equation:
$2x + 5y = -5y + 5y$
$2x + 5y = 0$
Step 2: Structuring the Standard Form
The standard form requires three distinct terms: an $x$-term, a $y$-term, and a constant term $c$. In our current equation, $2x + 5y = 0$, the constant term is absent. Mathematically, an absent term implies a coefficient or value of zero. We explicitly introduce zero as the constant term to perfectly match the $ax + by + c = 0$ template:
$2x + 5y + 0 = 0$
Step 3: Coefficient Extraction
By directly comparing our structured equation with the general form $ax + by + c = 0$, we can extract the corresponding real number coefficients:
- The coefficient of $x$ corresponds to $a$. Thus, $a = 2$.
- The coefficient of $y$ corresponds to $b$. Thus, $b = 5$.
- The constant term corresponds to $c$. Thus, $c = 0$.
Graphical Representation (Geometric Interpretation)
A linear equation in two variables geometrically represents a straight line on a Cartesian plane. Because the constant term $c = 0$, the line must pass through the origin $(0,0)$. We can verify this by finding two points that satisfy $2x + 5y = 0$:
- If $x = 0$, then $5y = 0 \implies y = 0$. Point: $(0, 0)$
- If $x = 5$, then $10 + 5y = 0 \implies 5y = -10 \implies y = -2$. Point: $(5, -2)$
- If $x = -5$, then $-10 + 5y = 0 \implies 5y = 10 \implies y = 2$. Point: $(-5, 2)$
Final Solution: The linear equation expressed in standard form is $2x + 5y + 0 = 0$. The corresponding coefficient values are $a = 2$, $b = 5$, and $c = 0$.
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(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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