Algebra
What this domain tests
Algebra covers linear equations and inequalities, systems of equations, slope and intercepts, exponent rules, factoring, and quadratic equations. The emphasis is on manipulating expressions correctly and interpreting what solutions mean.
Concepts you must know
- Linear form: y = mx + b, where m is slope (rise/run) and b is the y-intercept. Slope between two points is (y₂ − y₁)/(x₂ − x₁).
- Solving systems: substitution or elimination. A system with the same slope but different intercepts has no solution; same slope and same intercept has infinitely many.
- Inequalities: multiplying or dividing by a negative flips the inequality sign.
- Exponents: xᵃ · xᵇ = xᵃ⁺ᵇ; (xᵃ)ᵇ = xᵃᵇ; x⁰ = 1; x⁻ᵃ = 1/xᵃ.
- Quadratics: factor, or use x = (−b ± √(b² − 4ac))/(2a).
Traps
- Forgetting to flip the inequality sign on negative division.
- Distributing a negative incorrectly: −(x − 3) = −x + 3, not −x − 3.
- Treating (x + 3)² as x² + 9 instead of x² + 6x + 9.
- Plugging a solution back in-always verify in the original equation.
Worked example
Solve 3(x − 2) = 5x + 4. Distribute: 3x − 6 = 5x + 4. Subtract 3x: −6 = 2x + 4. Subtract 4: −10 = 2x. So x = −5. Check: 3(−7) = −21 and 5(−5) + 4 = −21. Correct.
Rule of thumb
Do the same operation to both sides, one step at a time, and substitute your answer back into the original equation to catch sign errors.