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. \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2} = \frac{c_1}{c_2}\)
• 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:

Remember the clear definitions:
- Consistent and Independent: Unique solution (Intersecting lines) \(\implies \frac{a_1}{a_2} \neq \frac{b_1}{b_2}\).
- Inconsistent: No solution (Parallel lines) \(\implies \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\).
- Consistent and Dependent: Infinitely many solutions (Coincident lines) \(\implies \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\).

Answer 1

The Correct Option is D

This question is testing the three cases that can happen when we have two linear equations in two variables: a unique solution, no solution, or infinitely many solutions. Rather than stating the rule and matching it, we can rule out each wrong option by thinking about what "consistent and dependent" actually means geometrically: the two lines must be the same line, drawn twice.

1. $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$: this describes two lines with different slopes, so they cross at exactly one point. That is a unique solution, not infinitely many, so this cannot be the dependent case.
2. $\frac{a_1}{a_2} \neq \frac{b_1}{b_2} = \frac{c_1}{c_2}$: the first part already says the slopes differ, so again the lines meet at one point only. This condition does not even make consistent sense together with "dependent", so it is wrong.
3. $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$: here the slopes match but the lines are shifted apart (different intercepts), so they run side by side and never touch. That is the no-solution, inconsistent case, the exact opposite of what the question wants.
4. $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$: here every ratio matches, which forces the second equation to just be a scaled copy of the first. Both equations then describe the exact same line, so every point on that line satisfies both equations at once. This gives infinitely many common solutions, which is exactly what "consistent and dependent" means.

Only the last option leaves the two equations describing one single overlapping line, so it is the only one that matches a consistent and dependent system.

Let's summarize:

• Different slope ratios always give a single intersection point, never infinitely many solutions.
• Same slope ratio but different constant ratio means parallel, non-touching lines.
• All three ratios equal means the lines coincide, which is the only way to get infinite shared solutions.

So the condition for a consistent and dependent pair of equations is $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, option (D).

\[ \boxed{\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}} \]