NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 3(i)
NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables
Question:
On comparing the ratios \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\), find out whether the following pair of linear equations are consistent or inconsistent.
\(3x + 2y = 5\) ; \(2x - 3y = 7\).
Given:
The pair of linear equations is:
\(3x + 2y = 5\)
\(2x - 3y = 7\)
To Find:
Whether the given pair of linear equations are consistent or inconsistent by comparing the ratios of coefficients.
Formula:
For two linear equations in the form:
\(a_1x + b_1y + c_1 = 0\)
\(a_2x + b_2y + c_2 = 0\)
The pair is:
- Consistent (unique solution) if \(\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}\)
- Inconsistent (no solution) if \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}\)
- Consistent (infinitely many solutions) if \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\)
Solution:
Step 1: Write the equations in standard form.
$$ \begin{align*} 3x + 2y &= 5 \implies 3x + 2y - 5 = 0 \\ 2x - 3y &= 7 \implies 2x - 3y - 7 = 0 \end{align*} $$Step 2: Identify the coefficients \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\).
$$ \begin{align*} a_1 = 3,\quad b_1 = 2,\quad c_1 = -5 \\ a_2 = 2,\quad b_2 = -3,\quad c_2 = -7 \end{align*} $$Step 3: Find the ratios of the coefficients.
$$ \frac{a_1}{a_2} = \frac{3}{2},\quad \frac{b_1}{b_2} = \frac{2}{-3},\quad \frac{c_1}{c_2} = \frac{-5}{-7} = \frac{5}{7} $$Step 4: Compare the ratios.
$$ \frac{a_1}{a_2} = \frac{3}{2},\quad \frac{b_1}{b_2} = \frac{2}{-3} $$Since \(\frac{3}{2} \neq \frac{2}{-3}\), the ratios are not equal.
Step 5: Conclusion.
Since \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), the given pair of equations is consistent and has a unique solution.
Result:
The given pair of linear equations is consistent (has a unique solution).
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