What is a linear equation?
A linear equation is one where the variable appears only to the first power, with no x², no x under a root and no x in a denominator. Examples are 3x − 7 = 11 and 5x + 9 = 2x − 12.
The name comes from graphing: y = 2x + 1 makes a straight line. When you solve a one-variable linear equation, you get exactly one answer, no answer, or every number, and nothing in between.
How to solve linear equations step by step
Most textbooks teach the same general strategy. It handles any linear equation, however messy.
- Simplify each side: distribute to clear parentheses and combine like terms.
- Collect the variable terms on one side by adding or subtracting.
- Collect the constant terms on the other side.
- Divide both sides by the coefficient of the variable.
- Check by substituting into the original equation.
If the equation has fractions, you can multiply through by the common denominator before step 1. That's optional, and the solver keeps fractions as they are.
Not every equation needs all five steps. 3x − 7 = 11 has no parentheses and x on only one side, so you skip straight to steps 3 and 4. The list is a checklist, and you tick off whatever applies.
How do you solve 5x + 9 = 2x − 12?
Subtract 2x from both sides, subtract 9 from both sides, then divide by 3. The answer is x = −7.
Moving the 2x to the left keeps the x coefficient positive (5 − 2 = 3). You could move 5x to the right instead and get −21 = −3x, which leads to the same x = −7 with one extra sign to watch.
What does a multi step equation calculator do differently?
Nothing different, just more of it. A multi-step equation is a linear equation that needs distributing or combining like terms before the usual two moves.
Try 3(2x − 1) = 4x + 9. Distribute to get 6x − 3 = 4x + 9. Subtract 4x for 2x − 3 = 9, add 3 for 2x = 12, and divide to get x = 6. The solver shows the distribution as its own line, which is where most errors start.
When does a linear equation have no solution or infinitely many?
If the x terms cancel and leave a false statement, there's no solution. If they cancel and leave a true statement, there are infinitely many solutions, meaning every real number works.
| Equation | After the x terms cancel | Result |
|---|---|---|
| 5x + 9 = 2x − 12 | 3x = −21 | One solution, x = −7 |
| 3(x + 2) = 3x + 1 | 0 = −5 | No solution |
| 3(x + 2) = 3x + 6 | 0 = 0 | All real numbers |
You can often spot these before solving. If both sides have the same x coefficient after simplifying, the answer is either none or all, depending on whether the constants match. In graph terms, no solution means two parallel lines and infinitely many means the same line twice.
Can the linear equation solver handle fractions and decimals?
Yes. Type fractions with a slash and decimals as usual. For x/2 + x/3 = 10 it combines the x terms into (5/6)x and returns x = 12. For 0.5x + 1.2 = 3.7 it returns x = 5.
Answers come back as exact fractions when they aren't whole numbers, so you won't see a rounded decimal where your teacher expects 7/3.
What are the most common mistakes?
Moving a term across the equals sign without changing its sign is the big one. "Move 2x to the left" means "subtract 2x from both sides", which is why 2x on the right becomes −2x on the left.
Another is distributing a negative to only the first term. In −2(x − 4), both terms flip: −2x + 8. Writing −2x − 8 changes the answer.
Finally, students often read 0 = 0 as "x = 0". It isn't. It means every value of x works.
How do you turn a word problem into a linear equation?
Pick a letter for the unknown, write each part of the sentence as an expression, and set the two sides equal. Then solve it like any other equation.
Say a phone plan costs $20 a month plus $0.10 per text, and this month's bill is $35. Let x be the number of texts. The cost is 20 + 0.1x, and it equals 35.
So 150 texts were sent. Check it in the story, not only the equation: 150 texts at 10 cents is $15, and $20 plus $15 is $35. A fixed amount plus a rate times a quantity is the most common shape for these problems, so it's worth spotting quickly.