3x + 4 = 3x + 9 has no solution, and that is the answer rather than a failure to find one. Subtract 3x from both sides and the unknown disappears from both, leaving 4 = 9. That statement is simply false, and no value of x can rescue it, because x is no longer in it. There is nothing left to choose....
Type the equation as it is written, brackets and all. Three answers are possible and all three are real: one value, no value, or every value. When the unknown cancels out that is not a failure — it is the answer, and which of the two it is depends on the constant left behind. Anything that is not linear is refused rather than fudged: x² and an unknown in a denominator both get a reason, not a number. Decimals are read as exact fractions, so 0.1 + 0.2 is 3/10 here rather than 0.30000000000000004.
Brackets, fractions, decimals and the unknown on both sides all work. Any single letter can be the unknown.
NO SOLUTION
no solution
The unknown cancels and leaves −5 = 0, which is false whatever x is.
| As typed | 3x + 4 = 3x + 9 |
| Tidied | 3x + 4 = 3x + 9 |
| All on one side | −5 = 0 |
COEFFICIENT OF X
0
it cancelled, which decides the answer
CONSTANT LEFT OVER
−5
what is left after moving across
THE TIDIED EQUATION
3x + 4 = 3x + 9
both sides expanded and collected
WHAT THAT MEANS
a contradiction
false for every value of the unknown
WHAT IS HAPPENING
3x + 4 = 3x + 9
Brackets multiplied out, numbers combined, and everything written as a multiple of the unknown plus a constant. Nothing has been moved across the equals sign yet — this is only tidying.
−5 = 0
Subtracting the right-hand side from the left. What matters now is the coefficient of the unknown: if it survives there is one solution, and if it cancels the answer depends entirely on the constant left behind.
−5 = 0
The unknown vanished from both sides, leaving a statement that is simply false. No value of x can make −5 equal zero, so the equation has no solution. The two sides describe parallel lines that never meet.
Why there are exactly three answers. Each side of a linear equation describes a straight line, and asking where they are equal is asking where the lines meet. Two straight lines can cross once, run parallel and never meet, or be the same line and meet everywhere. There is no fourth possibility, which is why an equation like this can never have exactly two solutions — and why the case that looks like a dead end, the one where the unknown vanishes, is really the graph telling you the lines were parallel or identical all along.
Exact fractions throughout · the answer is substituted into the equation as you typed it
Subtract 3x from both sides. The unknown disappears from both, which is the signal that this is one of the two unusual cases.
Read what is left: 4 = 9. It contains no unknown, so it is either true or false outright — and it is false.
Conclude that there is no solution. Nothing can be substituted, because no value of x appears in the contradiction.
Change the 9 to a 4 in the box above and watch the answer flip to every number — the same cancelling, the opposite outcome.
3x + 4 = 3x + 9 Subtract 3x from both sides: 3x + 4 − 3x = 3x + 9 − 3x 4 = 9 The unknown has gone from both sides at once, because both sides carried exactly the same 3x. What remains is a statement about numbers alone, and it is false: four is not nine. Since x does not appear in 4 = 9, no choice of x can make it true. The equation has no solution. Read it on a graph and it stops being mysterious. The left side is the line y = 3x + 4 and the right is y = 3x + 9. Both climb three units for every one across, so they are parallel, and one sits five above the other everywhere. Parallel lines never meet, so there is no x at which the two sides are equal. Now the neighbour. Change the 9 to a 4 and the equation becomes 3x + 4 = 3x + 4. The same subtraction leaves 0 = 0, which is true regardless of x, so every number is a solution — the two lines are the same line drawn twice. Those are the only three outcomes available: one crossing, none, or everywhere. Two straight lines cannot arrange themselves any other way.
| Rule | What it says | Why |
|---|---|---|
| What counts as linear | the unknown to the first power | No x², no x under a division, no x inside a root. |
| One solution | the x terms do not cancel | The usual case. Divide by what is left of the coefficient. |
| No solution | 2x + 1 = 2x + 3 | The x terms cancel and the constants disagree. Parallel lines. |
| Every number works | 2(x+1) = 2x + 2 | Both sides are the same expression. An identity, not an equation. |
| Cancelling is not failure | it is the answer | Which of the two answers depends on what is left behind. |
| Do the same to both sides | add, subtract, multiply, divide | Any of the four, as long as you never divide by zero. |
| Clearing fractions | multiply by the common denominator | Legitimate because it is done to both sides equally. |
| Expanding brackets | multiply every term inside | The commonest slip is missing the last term of a long bracket. |
| A negative outside a bracket | −(x − 3) = −x + 3 | Both signs flip. The second one is the one people miss. |
| Most answers are fractions | 71.3% of small cases | A whole-number answer is the exception, not the norm. |
| Decimals are not decimal | 0.1 + 0.2 ≠ 0.3 in binary | Typed decimals are read as exact fractions here. |
| Two letters is a system | x + y = 3 | One equation with two unknowns needs a second equation. |
| Check by substituting | put the answer back in | Into the original, not the tidied version — that is the real check. |
| The graph reading | each side is a line | The solution is where they cross; parallel means none, identical means all. |
Because subtracting 3x from both sides removes the unknown entirely and leaves 4 = 9, which is false. There is no x left in the statement, so no value can be chosen to make it true. The two sides increase at exactly the same rate while starting five apart, so they stay five apart for ever.
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Last updated: August 31, 2026 · Exact fractions throughout, never floating point · When the unknown cancels, that is the answer rather than a failure.