Question

If the pair of linear equations : \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) is consistent and dependent, then

• A. \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)
• B. \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)
• C. \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \)
• D. \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)

Hint:

Dependent lines are just the same line written differently. For example, \( x+y=2 \) and \( 2x+2y=4 \).

Answer 1

The Correct Option is B

To determine the condition for a pair of linear equations to be consistent and dependent, consider the given pair of equations:

\(a_1x + b_1y + c_1 = 0\)

\(a_2x + b_2y + c_2 = 0\)

For the pair of linear equations to be consistent and dependent, all lines represented by the equations should coincide with each other. This means that each equation is a multiple of the other and they do not just intersect at a single point but overlap completely.

The condition for consistency and dependency is given by:

\(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)

This implies that the coefficients of equivalent terms in both equations are proportional in such a way that they are equal.

Now, let's evaluate the given options:

1. \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\) - This option suggests that the coefficients are not proportional, which means the equations might intersect or be parallel, but not coincident. This indicates inconsistency or independence, not dependency.
2. \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) - This correctly represents the condition of consistent and fully dependent equations.
3. \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\) - Here, the constant terms \(c_1\) and \(c_2\) are not proportional, which implies the lines could be parallel but not coincident, potentially leading to inconsistency.
4. The last option repeats option 2.

Thus, the correct answer is:

\(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)

Therefore, the pair of equations is consistent and dependent only under this condition.