2x + 3 = 7 is the equation almost everyone meets first, and it is worth slowing down on because the shape of it repeats for years afterwards. There is one unknown, it is multiplied by something, and something else is added on. Everything after this is the same two moves wearing different numbers. T...
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 = 2
Both sides come to 7 at this value, checked against the equation as you typed it.
| As typed | 2x + 3 = 7 |
| Tidied | 2x + 3 = 7 |
| All on one side | 2x − 4 = 0 |
THE ANSWER PUT BACK INTO WHAT YOU TYPED
left side → 7
right side → 7 ✓
COEFFICIENT OF X
2
it survives, so there is one solution
CONSTANT LEFT OVER
−4
what is left after moving across
AS A DECIMAL
2
exact — the answer is a whole number
BOTH SIDES THERE
7
the two sides agree at the answer
WHAT IS HAPPENING
2x + 3 = 7
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.
2x − 4 = 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 = 4 ÷ 2 = 2
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 7 · right 7
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
Subtract 3 from both sides. The left becomes 2x and the right becomes 4, and the equation is still balanced because both sides lost the same amount.
Divide both sides by 2, giving x = 2. This is the second and last undoing, and it comes second because the multiplication was applied first.
Substitute back into the original: 2(2) + 3 = 4 + 3 = 7, which matches the right-hand side exactly.
Change the numbers in the box above to try a variant — the working updates as you type, and the two steps stay the same shape.
2x + 3 = 7 Both sides carry the same equation, so anything done to one has to be done to the other. Start by removing the +3: 2x + 3 − 3 = 7 − 3 2x = 4 Now the left is only a multiplication, and undoing a multiplication means dividing: 2x ÷ 2 = 4 ÷ 2 x = 2 Check it in the original, which is the only check worth making: 2(2) + 3 = 4 + 3 = 7. The right-hand side is 7, so the two agree. Why that order? The expression 2x + 3 does two things to x — multiplies by 2, then adds 3. Undoing runs in reverse, so the adding is undone first. Try it the other way and see what happens: dividing 2x + 3 = 7 by 2 straight away gives x + 1.5 = 3.5, which is still true and still leads to x = 2, but only if you divided the 3 as well. Dividing only the 2x term is the mistake, and it gives x + 3 = 3.5 and an answer of 0.5.
| 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 2x + 3 multiplies by 2 first and adds 3 second, so undoing goes backwards: the addition comes off before the multiplication. Dividing first is not wrong if you divide every term, but it introduces fractions early and it invites the real error, which is dividing 2x by 2 while leaving the 3 alone.
24 Times Table
Calculate instantly →
Quadratic Equation Solver
Calculate instantly →
Logarithm Calculator
Calculate instantly →
Fraction Calculator
Calculate instantly →
Fraction Simplifier
Calculate instantly →
Linear Equation Solver
Calculate instantly →
25 Times Table
Calculate instantly →
26 Times Table
Calculate instantly →
27 Times Table
Calculate instantly →
Simultaneous Equations Solver (2×2)
Calculate instantly →
28 Times Table
Calculate instantly →
Exponent Calculator
Calculate instantly →
29 Times Table
Calculate instantly →
30 Times Table
Calculate instantly →
Times Tables Mega Calculator
Calculate instantly →
Square Root of 10
Calculate instantly →
Square Root of 1
Calculate instantly →
Square Root Mega
Calculate instantly →
Square Root of 2
Calculate instantly →
23 Times Table
Calculate instantly →
Last updated: September 1, 2026 · Exact fractions throughout, never floating point · When the unknown cancels, that is the answer rather than a failure.