NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 2(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 lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:
\( 5x - 4y + 8 = 0 \)
\( 7x + 6y - 9 = 0 \)
Given:
The pair of linear equations:
\( 5x - 4y + 8 = 0 \)
\( 7x + 6y - 9 = 0 \)
To Find:
Whether the lines represented by the given equations:
- Intersect at a point
- Are parallel
- Are coincident
Formula:
For two linear equations in the form:
\( a_1x + b_1y + c_1 = 0 \)
\( a_2x + b_2y + c_2 = 0 \)
Compare the ratios:
\[
\frac{a_1}{a_2},\quad \frac{b_1}{b_2},\quad \frac{c_1}{c_2}
\]
Case 1: If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the lines intersect at a point.
Case 2: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), the lines are parallel.
Case 3: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \), the lines are coincident.
Solution:
Step 1: Write the equations in standard form.
\[ \begin{align*} 5x - 4y + 8 &= 0 \\ 7x + 6y - 9 &= 0 \end{align*} \]Step 2: Identify the coefficients.
\[ \begin{align*} a_1 = 5, \quad b_1 = -4, \quad c_1 = 8 \\ a_2 = 7, \quad b_2 = 6, \quad c_2 = -9 \end{align*} \]Step 3: Find the ratios of corresponding coefficients.
\[ \frac{a_1}{a_2} = \frac{5}{7} \] \[ \frac{b_1}{b_2} = \frac{-4}{6} = -\frac{2}{3} \] \[ \frac{c_1}{c_2} = \frac{8}{-9} = -\frac{8}{9} \]Step 4: Compare the ratios to determine the relationship between the lines.
\[ \frac{a_1}{a_2} = \frac{5}{7} \approx 0.714 \] \[ \frac{b_1}{b_2} = -\frac{2}{3} \approx -0.667 \] Since \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), by Case 1, the lines intersect at a single point.Result:
The lines represented by the equations \( 5x - 4y + 8 = 0 \) and \( 7x + 6y - 9 = 0 \) intersect at a single point.
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