Overview

Algebra in contests rewards clean manipulation: simplify, factor, and solve. Build habits like checking units, substituting smart values, and rewriting in equivalent forms. Pranav

Key Ideas

• Solve linear equations by isolating variables; keep track of constraints.
• Factor common patterns: $a^2 - b^2 = (a-b)(a+b)$ and $a^2 + 2ab + b^2$.
• Use substitution to reduce complex systems to a single variable.

Worked Example

Solve $\frac{3}{x} + \frac{2}{x+1} = 1$.

Multiply both sides by $x(x+1)$ to clear denominators: $3(x+1) + 2x = x(x+1).$ So $3x + 3 + 2x = x^2 + x$, giving $x^2 - 4x - 3 = 0$. Then $x = 2 \pm \sqrt{7}$. Check that neither solution makes a denominator $0$.

Common Pitfalls

• Clearing denominators but forgetting to track excluded values.
• Expanding when factoring is faster.

Practice Problems