Linear Equation Calculator
This linear equation calculator solves for x in one unknown, reads the equation exactly as your question writes it, and tells you which of the three possible answers you have.
Type 3(x − 4) = 2x + 7 and you get x = 19 — with both sides worked out at x = 19 to prove it.
Four Steps, Always the Same Four
Open the brackets. Gather the x terms on one side. Gather the numbers on the other. Divide by whatever is left in front of x.
That order matters. Moving terms across before the brackets are open is where most of the damage happens, because a minus sitting outside a bracket has to reach every term inside it. −(x − 3) becomes −x + 3, and the second sign is the one people forget.
Three Answers Are Possible, Not One
Most calculators assume you are heading for a number. Sometimes you are not, and knowing that is worth more than the arithmetic.
2x + 3 = 2x + 5. Take 2x off both sides. You are left with 3 = 5, which is false — and x is no longer in it, so nothing you choose can rescue it. No solution.
2(x + 1) = 2x + 2. Open the bracket and both sides read 2x + 2. They were always the same expression. Every number works. Infinitely many solutions, and the proper name for it is an identity.
Both of these are set questions from Year 8 onwards, and both get written down as "no answer" by students who think they have made a mistake. They have not. The answer is "no solution", and it earns full marks written that way.
| After the x terms cancel | What is left | The answer |
|---|---|---|
| x survives | ax = b | one solution |
| x vanishes, statement true | 0 = 0 | every number |
| x vanishes, statement false | 0 = 2 | no solution |
Why Fractions Should Stay Fractions
7x = 3 gives x = 3/7. Not 0.4286.
The fraction is the answer; the decimal is a rounded reading of it. Carry 0.4286 into the next line of a longer question and the error grows with every step. Most marking schemes want the exact form, and several will not accept the decimal at all.
Nothing here is ever converted to a decimal behind your back. Your equation is turned into whole-number fractions the instant it is read, and it stays that way through every step. x/3 + 1/4 = 5/6 comes back as 7/4, not 1.75.
Always Put It Back
The check costs twenty seconds and catches every sign error you could have made.
Take x = 19 back to 3(x − 4) = 2x + 7, the equation you started with rather than the tidied version. Left side: 3(19 − 4) = 3 × 15 = 45. Right side: 2 × 19 + 7 = 45. They match, so the answer is right and there is nothing left to worry about.
This page will not print an answer until that check has passed. For the other two outcomes it goes further: before calling an equation an identity it tries x = 0, x = 7 and x = −13/5 and requires all three to work, and before calling one a contradiction it requires all three to fail. Saying "no solution" wrongly is worse than saying nothing.
Worth Checking Your Source
Here is one worth doing by hand. 9(x − 1) − 35 = 8x + 37.
Open the bracket: 9x − 9 − 35 = 8x + 37, so 9x − 44 = 8x + 37, so x = 81.
Test it. Left: 9(81 − 1) − 35 = 720 − 35 = 685. Right: 8 × 81 + 37 = 648 + 37 = 685. Correct.
There is a widely shared worked example of this exact equation that finishes at x = 80. Substitute 80 and the two sides come out 676 and 677. The lesson is not that the internet is unreliable — it is that the check takes twenty seconds and would have caught it.
Clearing Fractions First, or Carrying Them Through
Two ways, and both are right.
Take x/3 + 1/4 = 5/6. You can multiply every single term by 12, the lowest number all three denominators divide into, which turns it into 4x + 3 = 10 and leaves nothing awkward to carry. Or you can work with the fractions as they stand and add them the usual way. Either route arrives at x = 7/4.
Clearing them first usually means fewer mistakes, for one reason: once the fractions are gone, so is the temptation to reach for a decimal halfway through. The moment 1/3 becomes 0.333 the answer stops being exact, and on a multi-mark question that is where the marks quietly go.
One warning about multiplying through. Every term gets multiplied, including the ones that look harmless. In x/3 + 1 = 5/6 the number 1 becomes 12, not 1 — forgetting the term without a denominator is the second most common error in this topic after the bracket sign.
How to Use This Linear Equation Calculator
Type the equation the way the question writes it, brackets and all: 3(x - 4) = 2x + 7. No rearranging first — the rearranging is the part this page is for. One equals sign, one letter x, and fractions or decimals wherever you like.
The answer appears with the check underneath it, and Show the working opens all four steps on your own numbers. If your question has two unknowns rather than one, that is a system of equations. If x is squared anywhere, you need the quadratic formula instead. An equation with x underneath a fraction line is a rational equation, and one with an inequality sign rather than an equals is a linear inequality.
Linear Equation Calculator FAQ
How do you solve a linear equation?
What does it mean when x disappears from the equation?
Can a linear equation have no solution?
Can a linear equation have infinitely many solutions?
Should I leave the answer as a fraction or a decimal?
How do I check my answer to a linear equation?
What do I do with fractions inside the equation?
Why does the minus sign in front of a bracket cause so many errors?
Open the brackets, move the x one way and the numbers the other, then put your answer back to be sure.
