Solve the Equation - Solution & Explanation

Problem Overview: The input is a string representing a linear equation such as x+5-3+x=6+x-2. Your job is to isolate x and return the solution in the form x=#value. If both sides reduce to the same expression, the result is Infinite solutions. If the equation becomes contradictory, return No solution.

Approach 1: Balancing Equation Terms (O(n) time, O(1) space)

Split the equation into left and right parts using the = sign. Parse each side character by character and track two values: the coefficient of x and the constant sum. When you encounter a term like 2x, update the coefficient; when you encounter a number like 5, update the constant. After processing both sides, move all x terms to one side and constants to the other by subtracting coefficients and constants appropriately.

This effectively “balances” the equation: ax + b = cx + d becomes (a - c)x = d - b. If the resulting coefficient of x is zero, check whether the constants are equal to determine infinite or no solutions. The implementation relies on careful string parsing and sign tracking, which makes it a practical exercise in string processing combined with simple math operations.

Approach 2: Combining and Simplifying Terms (O(n) time, O(1) space)

Instead of handling left and right sides separately, scan the entire equation once and normalize the terms while traversing. Track the current sign and switch behavior when crossing the = symbol. Terms before = contribute normally; terms after it contribute with inverted signs because they conceptually move to the left side.

For each token, determine whether it is an x term or a numeric constant. Maintain two accumulators: total coefficient of x and total constant value. For example, encountering +3x adds 3 to the coefficient, while -5 subtracts 5 from the constant. When the scan completes, the equation reduces to ax + b = 0. Solving becomes straightforward: compute x = -b / a, with special handling when a == 0.

This method treats the equation as a single stream of tokens and uses a lightweight simulation of algebraic term movement. It avoids storing intermediate lists of tokens and keeps the logic compact.

Recommended for interviews: Interviewers usually expect the linear scan that aggregates coefficients and constants. It demonstrates comfort with string parsing and algebraic simplification in O(n) time and O(1) space. Implementing the balanced two-side parsing first can help reason about the equation structure, but the single-pass simplification shows stronger problem-solving efficiency.