Master Linear Equations from Zero

A linear equation is an equation where the highest power of the variable is 1. Example: 2x + 3 = 7. In placement exams, linear equations appear as: 1. Word problems (competitive exam scoring) 2. Simultaneous equations (two unknowns, two equations) 3. Function-based equations (f(m+n) = f(m) + f(n) + ...) The key skill is translating English sentences into equations.

Key Formulas

Simple Linear

ax + b = 0 → x = -b/a

Isolate the variable by moving constants to one side.

Simultaneous Equations

a1x + b1y = c1 and a2x + b2y = c2

Multiply and subtract to eliminate one variable.

Exam Scoring Problems

Correct × marks_per_correct - Wrong × marks_per_wrong = Score

Classic placement exam problem. Two scenarios → two equations.

⚡ Shortcut Tricks

Exam scoring type: Two scoring schemes, same test → 2 equations, 2 unknowns (correct + wrong = total). Solve simultaneously.

Function type f(m+n) = f(m) + f(n) + something: Build from f(1) step by step using powers of 2.

Whenever 'find x+y' is asked (not x and y separately), look for a shortcut — add both equations.

Substitution is faster than elimination when one equation has a coefficient of 1.

For word problems: define variables FIRST. 'More than' = add. 'Less than' = subtract. 'Times' = multiply.

Check your answer by substituting back — saves time in MCQ exams.

✏️ Solved Examples

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