Algebra Foundations

One-Step Equations

Solving equations that need a single inverse operation.

Foundational15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

From Expressions to Equations

An expression like 2x + 1 can be evaluated or simplified, but it doesn't have one fixed value — it changes depending on what x is. An equation is different: it's a statement that two expressions are equal, like x + 5 = 12, and that statement is only true for specific values of x. Solving an equation means finding the value (or values) that make it true.

Definition — Solving an Equation

Finding the value of the variable that makes the equation a true statement. A one-step equation is one that can be solved using a single inverse operation.

The core idea behind solving any equation is that both sides must always stay equal — like a balanced scale. Whatever you do to one side, you must do to the other, or the scale tips and the equation is no longer true. To isolate the variable, you undo whatever operation is attached to it, using the inverse operation.

If the equation has...	Undo it with...
addition (x + 5 = 12)	subtraction
subtraction (x − 5 = 12)	addition
multiplication (5x = 20)	division
division (x ÷ 5 = 4)	multiplication

Undoing an operation with its inverse

Worked Example — Solving an Addition Equation

Solve x + 5 = 12. Since 5 is being added to x, undo it by subtracting 5 from both sides: x + 5 − 5 = 12 − 5, which simplifies to x = 7. Check the answer by substituting it back in: 7 + 5 = 12. True, so x = 7 is correct.

Worked Example — Solving a Multiplication Equation

Solve 6x = 42. Since x is being multiplied by 6, undo it by dividing both sides by 6: 6x ÷ 6 = 42 ÷ 6, which simplifies to x = 7. Checking: 6 × 7 = 42. True, so x = 7 is correct.

Equation Editor

Edit as raw LaTeX

sum(i, 1, n, i^2) = n(n+1)(2n+1)/6

Constants

Structures

Calculus & discrete math

Greek

\sum_{i=1}^{n} i^{2} = \frac{\operatorname{n}\left(n + 1\right) \cdot \left(2 \cdot n + 1\right)}{6}

Evaluate

Unknown function "n"

Tip

Always check your solution by substituting it back into the original equation. If both sides come out equal, you've solved it correctly — and if they don't, you'll catch the mistake before moving on.

Common Mistakes

Performing the inverse operation on only one side of the equation, like solving x + 5 = 12 by writing x = 12 − 5 without also removing the 5 from the left side conceptually.
Think of the equation as staying balanced: whatever you subtract, add, multiply, or divide, apply it to both sides so the two sides remain equal.
Using the wrong inverse operation, such as multiplying to undo an equation where the variable was multiplied, instead of dividing.
Addition and subtraction are inverses of each other, and multiplication and division are inverses of each other. Match the undo operation to the one actually being reversed.

Key Takeaways

An equation states that two expressions are equal; solving it means finding the value that makes that true.
Undo the operation attached to the variable using its inverse operation, applied to both sides.
Checking a solution by substitution confirms whether it's correct.

Summary

One-step equations are solved with a single inverse operation applied to both sides. Many real equations need more than one step to isolate the variable — that's exactly what comes next.

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