Algebraic Relationships
Algebraic Relationships covers translating words into expressions, solving linear equations and inequalities, slope and graphing, systems of two equations, exponent rules, and quadratic patterns. Expect multi-step items and word problems that hinge on one clean setup.
Core rules
- Distribute and combine like terms before isolating; any operation applied to one side must be applied to the other.
- Slope m = (y₂ − y₁)/(x₂ − x₁). In y = mx + b, m is slope and b is the y-intercept. Parallel lines share m; perpendicular slopes are negative reciprocals whose product is −1.
- Multiplying or dividing an inequality by a negative number reverses the inequality symbol.
- Exponent rules: xᵃ · xᵇ = xᵃ⁺ᵇ; (xᵃ)ᵇ = xᵃᵇ; xᵃ/xᵇ = xᵃ⁻ᵇ.
- A system's solution is the ordered pair satisfying both equations - graphically, the intersection point. Constants added to y = x² shift the parabola up; a negative leading coefficient opens it downward.
Worked example
Solve 3(x − 2) = 5x + 8. Distribute: 3x − 6 = 5x + 8. Subtract 3x from both sides: −6 = 2x + 8. Subtract 8: −14 = 2x. Divide by 2: x = −7. Check by substituting: 3(−7 − 2) = −27 and 5(−7) + 8 = −27. Both sides agree, so the solution is confirmed rather than merely computed.
Traps that cost marks
- Distributing only to the first term: 3(x − 2) is 3x − 6, not 3x − 2.
- Forgetting to flip ≤ to ≥ after dividing by a negative coefficient.
- Computing slope as run over rise, or dropping a negative sign from a coordinate subtraction.
- Dividing both sides by a variable and silently discarding the solution where that variable equals zero.