Solving Linear Equations, Explained
A step-by-step guide to solving for an unknown in a linear equation — isolating the variable, balancing both sides, clearing fractions, and checking your answer — the foundation of all of algebra.
Updated 2026-06-04
What a linear equation is
A linear equation is a statement that two expressions are equal, where the unknown (usually written x) appears only to the first power — no squares, no roots, no variables multiplied together. Examples look like 3x + 5 = 20 or 2(x − 4) = x + 1. Solving one means finding the single value of x that makes the statement true.
The core idea is balance. An equation is like a scale: whatever you do to one side you must do to the other to keep it level. Every technique below is just a disciplined way of using that rule to peel everything away from the variable until it stands alone.
The tools you balance with
Inverse operations undo what is happening to the variable.
Addition and subtraction
Move a constant to the other side by adding or subtracting it from both sides.
Multiplication and division
Remove a coefficient by dividing both sides by it (or multiply to clear a fraction).
The distributive property
Expand expressions like 2(x − 4) into 2x − 8 before combining terms.
Combining like terms
Simplify each side by collecting all the x-terms and all the constants together first.
The solving procedure
Worked on 3x + 5 = 20.
Simplify each side
Distribute and combine like terms so each side is as clean as possible. Here both sides are already simple.
Move the variable to one side
Use addition or subtraction to get all x-terms on one side and constants on the other. Subtract 5: 3x = 15.
Isolate the variable
Divide both sides by the coefficient. Divide by 3: x = 5.
Check your answer
Substitute back into the original equation: 3(5) + 5 = 20. It holds, so the solution is correct.
Common questions
What if there are fractions?
Multiply every term on both sides by the least common denominator to clear the fractions, then solve the simpler equation as usual.
What does 'no solution' mean?
If the variables cancel and you are left with a false statement like 3 = 7, the equation has no solution. If you get a true statement like 5 = 5, every number is a solution.
Why check the answer?
Substituting your solution back into the original equation catches arithmetic slips instantly — it is the fastest way to know you are right.
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