NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 3(ii)
NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables
Question:
On comparing the ratios a1, b1, c1 and a2, b2, c2, find out whether the following pair of linear equations are consistent or inconsistent:
\(2x - 3y = 8;\quad 4x - 6y = 9.\)
Given:
- \(2x - 3y = 8\)
- \(4x - 6y = 9\)
To Find:
Whether the given pair of linear equations is consistent or inconsistent by comparing the ratios of coefficients.
Formula:
For a pair of linear equations:
\(a_1x + b_1y + c_1 = 0\)
\(a_2x + b_2y + c_2 = 0\)
- Consistent & Unique Solution: if \(\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}\)
- Consistent & Infinitely Many Solutions: if \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\)
- Inconsistent: if \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}\)
Solution:
Step 1: Write both equations in standard form.
$$ \begin{align*} 2x - 3y &= 8 \implies 2x - 3y - 8 = 0 \\ 4x - 6y &= 9 \implies 4x - 6y - 9 = 0 \end{align*} $$Step 2: Identify the coefficients \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\).
$$ \begin{align*} a_1 = 2,\quad b_1 = -3,\quad c_1 = -8 \\ a_2 = 4,\quad b_2 = -6,\quad c_2 = -9 \end{align*} $$Step 3: Find the ratios of the corresponding coefficients.
$$ \frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2} $$ $$ \frac{b_1}{b_2} = \frac{-3}{-6} = \frac{1}{2} $$ $$ \frac{c_1}{c_2} = \frac{-8}{-9} = \frac{8}{9} $$Step 4: Compare the ratios to determine consistency.
Since \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{1}{2} \) but \( \dfrac{c_1}{c_2} = \dfrac{8}{9} \),
i.e., \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2} \),
the system is inconsistent.
Result:
The given pair of linear equations is inconsistent. That means, there is no solution to this system of equations.
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