Linear Equations Guide for Beginners
Solve one-step and two-step linear equations with plain-language examples.
Quick Answer
ax + b = cx + d → (a − c)x = d − bWhen to Use This
When x appears on both sides, first collect the variable terms. Unlike a two-step equation with x on only one side, this can cause the entire variable term to disappear. That cancellation determines whether there is one solution, no solution or infinitely many.
Step-by-Step Method
- Subtract the right-side x term from both sides.
- Subtract the left-side constant from both sides.
- If the remaining coefficient is nonzero, divide to isolate x.
- If the coefficient is zero, compare the constants instead of dividing.
Calculator Tips and Limits
- Enter a simplified linear equation with x terms and one equals sign in the linked solver.
- A calculator cannot make division by zero valid; classify the remaining equality first.
Worked Examples
Variables on both sides
5x - 4 = 2x + 11 becomes 3x - 4 = 11 after subtracting 2x. Add 4 to get 3x = 15, then divide by 3 to get x = 5. Both original sides equal 21.
Cancellation reveals the solution set
2x + 3 = 2x + 3 simplifies to 3 = 3, so every real x works. In contrast, 2x + 3 = 2x + 4 becomes 3 = 4, which is false for every x.
Common Mistakes
Always expecting one number
Cancellation may yield an identity or contradiction instead of a single solution.
Calling 0/0 a solution
When the variable coefficient vanishes, return to the equality. The fraction 0/0 is not a valid way to identify x.
FAQ
What makes an equation linear?
After simplification, its variable terms have degree at most one. An x² term makes it quadratic unless it cancels.
Can I move a term by changing its sign?
That shorthand stands for adding or subtracting the same term on both sides. Writing that operation explicitly helps prevent sign errors.
References and corrections
OpenStax: Algebra and Trigonometry 2e — further reading. The worked examples on this page are CalcSolver explanations. Read our calculation and correction method, or report an error with the input and expected result.