Solving Linear Equations

Use structure before arithmetic

A linear equation has the variable raised only to the first power. Before moving terms, distribute parentheses and combine like terms on each side. This reduces noise and helps you avoid moving the wrong terms.

If denominators are simple and common, multiplying both sides by the least common denominator can remove fractions. That often makes the equation faster to solve and easier to check.

Recognize one solution, no solution, or infinitely many

Some equations simplify to a true statement such as 5 = 5. That means infinitely many solutions. Others simplify to a false statement such as 3 = 8. That means no solution.

• One solution: final form x = number.
• No solution: contradiction after simplification.
• Infinitely many: identity after simplification.

Solve an equation with fractions

A student solves (x/3) + 2 = (5x/6) - 1 during homework review.

1. Multiply both sides by 6 to clear denominators: 2x + 12 = 5x - 6.
2. Move x terms to one side: 12 = 3x - 6.
3. Add 6 to both sides: 18 = 3x, then divide by 3: x = 6.
4. Check in the original equation to confirm both sides are equal.

Common mistakes

• Adding a term to one side but forgetting to add it to the other side.
• Combining unlike terms such as 3x + 2 into 5x.
• Dropping negative signs when moving terms.
• Skipping the substitution check and missing arithmetic errors.

Try one

Sources

• OpenStax: College AlgebraOpen textbook with full chapter coverage, worked examples, and practice problems.
• Khan Academy: Algebra and Algebra 2Official lessons and practice sets for core algebra skills.