3x − 5 = 10 differs from the friendlier 3x + 5 = 10 in exactly one way, and that difference is where the marks go missing. The constant is being subtracted, so undoing it means adding — and the instinct to "do the opposite" often gets applied to the wrong thing, leaving 3x = 5 instead of 3x = 15. A...
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.
ONE SOLUTION
x = 5
Both sides come to 10 at this value, checked against the equation as you typed it.
| As typed | 3x - 5 = 10 |
| Tidied | 3x − 5 = 10 |
| All on one side | 3x − 15 = 0 |
THE ANSWER PUT BACK INTO WHAT YOU TYPED
left side → 10
right side → 10 ✓
COEFFICIENT OF X
3
it survives, so there is one solution
CONSTANT LEFT OVER
−15
what is left after moving across
AS A DECIMAL
5
exact — the answer is a whole number
BOTH SIDES THERE
10
the two sides agree at the answer
WHAT IS HAPPENING
3x − 5 = 10
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.
3x − 15 = 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.
x = 15 ÷ 3 = 5
The coefficient of the unknown is not zero, so there is exactly one value that works. Kept as a fraction, because most solutions are not whole numbers and a rounded one does not satisfy the equation.
left 10 · right 10
Both sides are evaluated at the answer from the expressions as they were typed, not from the tidied version. If those two numbers differ, something went wrong earlier and this is where it shows.
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
Add 5 to both sides. The left loses its −5 and becomes 3x; the right becomes 15. Adding is the opposite of subtracting, which is why it undoes it.
Divide both sides by 3, giving x = 5. The coefficient divides 15 exactly, so no fraction appears.
Check in the original: 3(5) − 5 = 15 − 5 = 10, which is the right-hand side.
Try changing the −5 to +5 in the box above and watch the answer drop from 5 to 5/3 — the same two steps, a different constant.
3x − 5 = 10 Read the left side as 3x plus negative five. To remove a negative five, add five, and do it to both sides: 3x − 5 + 5 = 10 + 5 3x = 15 Now undo the multiplication: 3x ÷ 3 = 15 ÷ 3 x = 5 Check: 3(5) − 5 = 15 − 5 = 10 ✓ The near-miss to watch for is subtracting the 5 instead of adding it, which gives 3x = 5 and then x = 5/3. That answer is wrong but not obviously so, and it survives a careless check if you substitute into 3x = 5 rather than into the original equation. Substituting 5/3 into 3x − 5 gives 5 − 5 = 0, nothing like 10, which is exactly the kind of disagreement a proper check is for. Compare it with 3x + 5 = 10, where the same method gives 3x = 5 and x = 5/3. Same coefficients, one sign different, and an answer that goes from whole to fractional.
| 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 adding is what undoes subtracting. The left side currently has five taken off it, so putting five back leaves 3x alone — and the right side must gain five as well to keep the equation balanced. Subtracting five instead makes the left side 3x − 10 and moves you further from the answer.
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Last updated: August 30, 2026 · Exact fractions throughout, never floating point · When the unknown cancels, that is the answer rather than a failure.