In short
You were solving a one-variable equation, you simplified honestly, and the last line came out as 0 = 0. That is not a mistake and it is not "no answer". It means every value of x satisfies the original equation — there are infinitely many solutions. The two sides of the original equation were really the same expression in disguise; once you peeled away the brackets and the constants, the disguise fell off and you saw the truth: 0 = 0, which is true for any x.
You're solving a textbook problem on the school bus and the equation looks ordinary: 2(x + 3) = 2x + 6. You distribute, you transpose, you cancel, and the last line stares back at you: 0 = 0. No x anywhere. No fraction, no decimal, no clean answer like x = 5. Your first instinct is panic — did I lose the variable somewhere? Your second instinct is to start over.
Don't. The equation is telling you something important. You did everything right. The reason there is no x in the final line is that the equation was never asking you to find a special x — it was an identity, a sentence that is true for every x you can throw at it. Plug in x = 0, plug in x = 7, plug in x = -100, plug in x = \pi — they all work. That is what "infinitely many solutions" means.
This is one of three things a one-variable equation can do when you finish solving it: hand you a unique x, or a contradiction like 0 = 5 (no solution — the sibling article covers that case), or this case — 0 = 0, every x works. Knowing all three outcomes is a quiet superpower in higher classes, because the same trichotomy returns in matrices, in systems of equations, in differential equations. Get comfortable with it now.
What "0 = 0" means
When you solve an equation, every legal step preserves the equality. Add the same number to both sides — equality holds. Multiply both sides by the same nonzero number — equality holds. Distribute brackets — the value of each side is unchanged. So the chain of equations you write down is really one statement, rewritten in different costumes.
Now, the final costume — the simplest possible form — is 0 = 0. Why: 0 = 0 is what mathematicians call a tautology — a sentence that is true regardless of the value of any variable, because no variable even appears. It is true on Monday, it is true on Diwali, it is true if x = 5 and it is true if x = -8000.
Run this backwards. If the simplified form is always true, then the original form must also be always true (because the two forms have exactly the same value for every x). So the original equation is satisfied by every real number. That is what we mean by "infinitely many solutions" in the one-variable setting.
There is a subtle vocabulary distinction worth knowing. Why: an equation is a sentence about x that might be true for some x and false for others — like 3x + 1 = 10, true only when x = 3. An identity is a sentence that is true for every x — like 2(x + 3) = 2x + 6. When you solve and reach 0 = 0, you have discovered that what looked like an equation was really an identity in disguise.
A concrete example
Take the equation
Distribute the 2 on the left:
Already you can see it — both sides are the same expression. But let's continue mechanically. Subtract 2x from both sides:
Subtract 6 from both sides:
There is the famous final line. No x, no work to do. The equation is true for every x.
Try it. Pick x = 4: left side = 2(4 + 3) = 14, right side = 2(4) + 6 = 14. Equal. Pick x = -10: left side = 2(-7) = -14, right side = -20 + 6 = -14. Equal. Pick x = 0: left side = 6, right side = 6. Equal. Pick whatever you want. The equation is an identity — the two sides are the same algebraic object, just written with and without the bracket.
The geometric picture
Whenever you have an equation \text{LHS}(x) = \text{RHS}(x), you can imagine plotting the two graphs y = \text{LHS}(x) and y = \text{RHS}(x) and asking where they cross. The x-coordinates of the crossing points are the solutions.
For 2(x + 3) = 2x + 6, the two graphs are
These are the same line. Same slope (2), same y-intercept (6). One line drawn on top of the other. They "cross" at every single point — which means every x on the number line is a solution.
Compare this to the three possible pictures.
So when you reach 0 = 0, you can think of it as the algebra telling you: the two graphs you were comparing are the same graph. Of course they "cross" everywhere.
Example 1: $4(x - 1) = 4x - 4$
Distribute the 4 on the left:
Why: the two sides are now visibly identical — same coefficient of x, same constant. The bracket on the left was hiding this fact.
Subtract 4x from both sides:
Add 4 to both sides:
Conclusion: infinitely many solutions. Every real number satisfies 4(x - 1) = 4x - 4. The original equation was an identity — the distributive law in costume.
Example 2: Verify by plugging in
Take the same equation, 4(x - 1) = 4x - 4. Pick four random values of x and check both sides.
| x | Left: 4(x - 1) | Right: 4x - 4 | Equal? |
|---|---|---|---|
| 0 | 4(-1) = -4 | 0 - 4 = -4 | yes |
| 1 | 4(0) = 0 | 4 - 4 = 0 | yes |
| 5 | 4(4) = 16 | 20 - 4 = 16 | yes |
| -3 | 4(-4) = -16 | -12 - 4 = -16 | yes |
Every value works. Why: this is what "infinitely many solutions" looks like in practice. You cannot list them all, but you can verify the pattern by sampling — and you can prove it for all x by simplifying both sides to the same expression, which is exactly what reaching 0 = 0 does.
Example 3: The contrast — change one number and the equation breaks
Now look at the close cousin: 4(x - 1) = 4x - 5. Just one digit different on the right.
Distribute the left: 4x - 4 = 4x - 5. Subtract 4x from both sides: -4 = -5, which simplifies to 0 = -1.
That is not 0 = 0. It is a contradiction — a sentence that is never true regardless of x. So the equation has no solution. The two graphs y = 4(x - 1) and y = 4x - 5 are parallel lines (same slope, different intercepts) that never meet.
Why: this is the "no solution" branch of the trichotomy. Compare carefully: changing the right side from 4x - 4 (identical to the left) to 4x - 5 (parallel but offset) flips the equation from "every x works" to "no x works". The siblings are mirror images — both lose the x when you simplify, but one ends in a tautology and the other in a contradiction.
How to write the answer in your exam
When the last line is 0 = 0, do not leave the answer blank or write "no solution" by mistake. Write something like:
Since both sides simplify to the same expression, the equation is an identity. The solution set is all real numbers — there are infinitely many solutions.
In set notation, you can write the solution set as \mathbb{R} or \{x \in \mathbb{R}\} or simply "x is any real number". CBSE and ICSE markers accept any clear phrasing.
Why this case matters in higher classes
The trichotomy you just learned — unique solution vs no solution vs infinite solutions — is not an algebra-class curiosity. It is the foundation for almost everything that comes next.
- In Class 10, when you solve systems of two linear equations, you classify each system as having a unique solution (intersecting lines), no solution (parallel lines), or infinite solutions (coincident lines) — exactly the same three cases.
- In Class 12 and JEE, the same trichotomy returns when you classify systems of n linear equations using matrices and determinants — "consistent and unique", "inconsistent", "consistent and dependent" are the same three labels in heavier vocabulary.
- In differential equations and linear algebra (college-level), the kernel and rank of a linear map are abstract names for exactly this question: does the equation have one solution, none, or a whole subspace of them?
Each time, the test for "infinitely many solutions" is the same: the two sides describe the same object. In one-variable algebra, that object is a number. In two-variable algebra, it is a line. In matrix algebra, it is a row of a matrix. Same idea, dressed differently.
A quick mental check
Whenever you finish solving a one-variable equation and the variable has vanished, ask one question: is the leftover sentence true or false?
- 0 = 0, 5 = 5, -3 = -3 — true. Infinitely many solutions (every x works).
- 0 = 5, 3 = 7, -1 = 1 — false. No solution (no x works).
That single check decides the answer. No need to redo the algebra. The variable disappeared because the original equation never depended on x in a meaningful way — it was either an identity (always true) or a contradiction (never true).
References
- NCERT, Mathematics Textbook for Class VIII, Chapter 2 (Linear Equations in One Variable). NCERT eTextbook.
- Wikipedia, Identity (mathematics) — the formal distinction between an equation and an identity.
- Wikipedia, Tautology (logic) — why 0 = 0 counts as a logically empty truth.
- Khan Academy, Number of solutions to linear equations — a video walkthrough of the same trichotomy.