In short
Dependent and inconsistent sound like cousins, but they sit at opposite ends of the solution scale. A dependent system has infinitely many solutions — the two equations describe the same line, so every point on that line works. An inconsistent system has zero solutions — the two equations describe parallel lines that never touch, so no point can satisfy both. After elimination, the leftover gives it away: 0 = 0 means dependent, 0 = c (with c \ne 0) means inconsistent. One says "everything works," the other says "nothing works." Don't conflate them.
You are revising for the CBSE Class 10 board exam. The question reads: "For what value of k is the system inconsistent?" You confidently write down the condition for infinite solutions, hand in the paper, and lose two marks because you mixed up two words that look like they should mean similar things. They don't. They mean opposites.
This article exists to make sure that never happens to you again.
Why these two words trick everyone
Look at the words honestly. Dependent sounds vaguely negative — "the equations depend on each other," like one is leaning on the other for support. Inconsistent also sounds negative — "they don't agree." Both seem to describe systems that are somehow broken. So your brain files them together in the "weird systems" drawer and stops paying attention to which is which.
That mental shortcut is wrong. Why: "dependent" and "inconsistent" point in opposite directions on the solution count. Dependent means the second equation is too friendly — it agrees so completely with the first that it adds no new information, and you get infinitely many solutions. Inconsistent means the second equation is too hostile — it contradicts the first, and you get zero solutions. The words sound similar; the meanings are extremes.
Read this line until it sticks:
Dependent = infinite solutions. Inconsistent = no solution. They are opposite ends of the same scale.
The comparison table
| Dependent (consistent dependent) | Inconsistent | |
|---|---|---|
| Number of solutions | Infinitely many (\infty) | Zero (0) |
| Geometry | Both equations are the same line (coincident) | Two parallel lines, never meet |
| Relation between equations | Eq. 2 is a scalar multiple of Eq. 1 — same information twice | Left sides are scalar multiples but right sides are not — same slope, different intercept |
| Elimination leftover | 0 = 0 (always true) | 0 = c where c \ne 0 (never true) |
| Coefficient ratios | \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} | \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2} |
| What the system "says" | "Every point on this line is a solution" | "No point in the plane works" |
Stare at the bottom two rows. The only difference between the dependent test and the inconsistent test is whether the third ratio joins the party. Why: if the left-hand-side coefficients are in the same ratio, the two lines have the same slope — they are parallel in some sense. If the constant on the right matches that ratio too, the lines have the same intercept and are actually the same line. If the constant breaks the pattern, the lines stay parallel but sit at different heights and never cross.
Three worked examples
Example 1 — Dependent: infinite solutions
Consider the system
To eliminate x, multiply equation 1 by 2:
Subtract this from equation 2:
The leftover is 0 = 0 — a statement that is always true, no matter what x and y are. Why: equation 2 is just 2 \times equation 1. They are the same line wearing two different outfits. So every point that satisfies the first equation automatically satisfies the second, and there are infinitely many such points — the entire line 2x + 3y = 6.
Verdict: dependent. Infinite solutions. A few examples that work: (0, 2), (3, 0), (6, -2), (-3, 4), ... pick any x, set y = (6 - 2x)/3, and you get a solution.
Example 2 — Inconsistent: no solution
Now consider
Same first move: multiply equation 1 by 2:
Subtract this from equation 2:
The leftover is 0 = 3 — a statement that is never true, no matter what x and y are. Why: the left-hand sides 4x + 6y are identical, but the right-hand sides demand two different values (12 from doubling equation 1, and 15 from equation 2 directly). One quantity cannot equal both 12 and 15 simultaneously. The two lines have the same slope -2/3 but different intercepts — parallel, never meeting.
Verdict: inconsistent. No solution. No matter how cleverly you pick x and y, you cannot make 4x + 6y equal both 12 and 15 at the same time.
Example 3 — The visual contrast, side by side
The two panels start with the same first equation and almost the same second equation. The only difference is the constant on the right of equation 2 — 12 versus 15. That single number flips the system from "every point on the line works" to "no point in the plane works." Why: the left-hand side 4x + 6y is determined by the slope. With matching slopes, the two equations are arguing only about the constant. If the constants agree (after the scaling that matched the slopes), the lines are identical. If they disagree, the lines are parallel but offset — and parallel lines never meet.
How to never confuse them again
Whenever you finish elimination on a 2 \times 2 system, look at the leftover:
- If you get a clean x = something or y = something, the system is independent — one unique solution. Continue back-substituting as normal.
- If the variables vanish and you are left with \mathbf{0 = 0}, the system is dependent — infinitely many solutions. Every point on the surviving line works.
- If the variables vanish and you are left with \mathbf{0 = c} for some nonzero c (like 0 = 3 or 0 = -7), the system is inconsistent — no solution at all. Stop and write "no solution."
This three-line decision rule is the entire skill. It works on every 2 \times 2 linear system you will ever meet in CBSE Class 10, and the same logic generalises to bigger systems in higher classes.
A quick mental anchor: in 0 = 0, the right side is zero, meaning "no contradiction" — anything is allowed, so infinite solutions. In 0 = c with c \ne 0, the right side is nonzero, meaning "the equations are contradicting each other by an amount of c" — nothing is allowed, so no solution. The right-hand side is the tell.
Where students lose marks
CBSE Class 10 graders are unforgiving on vocabulary. The board paper marking scheme distinguishes between "no solution" and "infinitely many solutions" word-for-word. Some traps to dodge:
- "Inconsistent" never means "infinite." If you write "the system is inconsistent and has infinitely many solutions," you have written a contradiction in your own answer, and you will lose the conceptual mark even if your ratio working is right.
- "Dependent" never means "no solution." A dependent system is consistent. The full name is "consistent dependent."
- A system with 0 = 0 leftover is not an error. Some students panic, scrub it out, and write "no solution." Wrong direction — 0 = 0 is the green light for infinitely many.
- A system with 0 = 5 (or any nonzero) leftover is not "x = 0, y = 5." It is "no solution." See the sibling article Got 0 = 5 from a System? That's "No Solution" for the deeper version of that trap.
For the broader vocabulary picture (consistent vs inconsistent and the sub-classification into independent vs dependent), see What does "consistent" vs "inconsistent" actually mean?. For the parent article on the four solution methods, see Systems of Linear Equations.