Solving linear equations in one variable — step by step algorithm

Question

Solve $\frac{3x + 5}{2} - \frac{x - 1}{3} = 4$ step by step. What is the general algorithm for solving any linear equation in one variable?

(CBSE 7-8 Board pattern)

Solution — Step by Step

flowchart TD
    A["Given linear equation"] --> B["Step 1: Clear fractions - multiply by LCM"]
    B --> C["Step 2: Expand all brackets"]
    C --> D["Step 3: Move variables to LHS, constants to RHS"]
    D --> E["Step 4: Combine like terms"]
    E --> F["Step 5: Divide both sides by coefficient of x"]
    F --> G["Step 6: Verify by substituting back"]
    G --> H["Final Answer"]

Why This Works

A linear equation is a balance — what is on the left equals what is on the right. Every operation we do (multiply both sides, add to both sides, divide both sides) preserves this balance. The strategy is to isolate $x$ by “undoing” everything around it.

Clearing fractions first makes the arithmetic cleaner — working with whole numbers is faster and less error-prone than juggling fractions throughout. Expanding brackets and collecting like terms reduces the equation to the simple form $ax = b$, which gives $x = b/a$.

Alternative Method

For simpler equations, use transposition directly — when a term moves from one side to the other, its sign changes:

$3x + 5 = 17$ becomes $3x = 17 - 5 = 12$, so $x = 4$.

Transposition is just a shortcut for “subtract from both sides.”

Common Mistake