NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 2(iii)

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 2(iii)

Question:

On comparing the ratios a1, b1, c1 and a2, b2, c2, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:

\(6x - 3y + 10 = 0\)
\(2x - y + 9 = 0\)

Given:

The two linear equations are:
1) \(6x - 3y + 10 = 0\)
2) \(2x - y + 9 = 0\)

To Find:

Whether the lines represented by these equations:

  • intersect at a point
  • are parallel
  • or are coincident
by comparing the ratios \( \frac{a_1}{a_2}, \frac{b_1}{b_2}, \frac{c_1}{c_2} \).

Formula:

For a pair of linear equations in the form:
\( a_1x + b_1y + c_1 = 0 \)
\( a_2x + b_2y + c_2 = 0 \)

The nature of the lines is determined by comparing the ratios:

  • \( \frac{a_1}{a_2}, \frac{b_1}{b_2}, \frac{c_1}{c_2} \)
Case 1: If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the lines intersect at a unique 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 and identify coefficients.

$$ \begin{align*} \text{Equation 1:} & \quad 6x - 3y + 10 = 0 \\ \text{Equation 2:} & \quad 2x - y + 9 = 0 \\ \end{align*} $$ Comparing with \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \), we get:
\( a_1 = 6, \; b_1 = -3, \; c_1 = 10 \)
\( a_2 = 2, \; b_2 = -1, \; c_2 = 9 \)

Step 2: Find the ratios \( \frac{a_1}{a_2}, \frac{b_1}{b_2}, \frac{c_1}{c_2} \).

$$ \frac{a_1}{a_2} = \frac{6}{2} = 3 \\ \frac{b_1}{b_2} = \frac{-3}{-1} = 3 \\ \frac{c_1}{c_2} = \frac{10}{9} $$

Step 3: Compare the ratios to determine the relationship between the lines.

$$ \frac{a_1}{a_2} = \frac{b_1}{b_2} = 3 \\ \frac{c_1}{c_2} = \frac{10}{9} \neq 3 $$ So, \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \).

Step 4: Conclusion.

According to the formula, this is Case 2.
Therefore, the lines are parallel and do not intersect.

Result:

The lines represented by the equations \(6x - 3y + 10 = 0\) and \(2x - y + 9 = 0\) are parallel and do not intersect at any point.

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