NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 2(ii)
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 lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:
\( 9x + 3y + 12 = 0 \)
\( 18x + 6y + 24 = 0 \)
Given:
The two linear equations are:
\( 9x + 3y + 12 = 0 \)
\( 18x + 6y + 24 = 0 \)
To Find:
Whether the lines represented by these equations:
(a) intersect at a point,
(b) are parallel,
or (c) are coincident.
Formula:
For two linear equations:
\( a_1x + b_1y + c_1 = 0 \)
\( a_2x + b_2y + c_2 = 0 \)
The nature of their graphs is determined by the ratios:
- Case 1: If \( \dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2} \), the lines intersect at a point (unique solution).
- Case 2: If \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2} \), the lines are parallel (no solution).
- Case 3: If \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} \), the lines are coincident (infinitely many solutions).
Solution:
Step 1: Write equations in standard form.
The given equations are already in standard form:
\( 9x + 3y + 12 = 0 \)
\( 18x + 6y + 24 = 0 \)
Step 2: Identify coefficients.
For the first equation:
\( a_1 = 9,\quad b_1 = 3,\quad c_1 = 12 \)
For the second equation:
\( a_2 = 18,\quad b_2 = 6,\quad c_2 = 24 \)
Step 3: Find the ratios \( \dfrac{a_1}{a_2} \), \( \dfrac{b_1}{b_2} \), \( \dfrac{c_1}{c_2} \).
\[ \dfrac{a_1}{a_2} = \dfrac{9}{18} = \dfrac{1}{2} \] \[ \dfrac{b_1}{b_2} = \dfrac{3}{6} = \dfrac{1}{2} \] \[ \dfrac{c_1}{c_2} = \dfrac{12}{24} = \dfrac{1}{2} \]
Step 4: Compare the ratios.
All three ratios are equal: \[ \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} = \dfrac{1}{2} \] Therefore, by Case 3, the lines are coincident.
Result:
The lines represented by \( 9x + 3y + 12 = 0 \) and \( 18x + 6y + 24 = 0 \) are coincident, i.e., they are the same line and have infinitely many solutions.
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