Problem-Solving

Direct relationship: both variables increase/decrease together (F=ma). Inverse relationship: one increases as the other decreases (Boyle's Law, PV=k).
Dimensional analysis treats units as algebraic quantities that multiply/divide/cancel — chain conversion factors so unwanted units cancel out.
Temperature conversions: F = (9/5)C + 32; K = C + 273.
Algebraic systems (2+ equations, multiple unknowns) solve via substitution, elimination, or setting equations equal when both are solved for the same variable.

Direct	Inverse
One variable increases, the other increases proportionally	One variable increases, the other decreases proportionally
e.g., F = ma (constant a: double m → double F)	e.g., PV = k (constant T: increase P → decrease V)

F = (9/5)C + 32, K = C + 273

1Write the starting quantity as a fraction with its units.
2Multiply by conversion factors (as fractions equal to 1) chosen so unwanted units cancel diagonally.
3Chain as many conversion factors as needed until only the target units remain.
4Example: 12 cm/s × (1 m/100 cm) × (60 s/1 min) = 7.2 m/min — cm cancels with cm, s cancels with s.

Method	When to use
Substitution	Isolate one variable in one equation, plug into the other
Elimination	Add/subtract equations (after matching coefficients) to cancel a variable
Setting equal	Both equations already solved for the same variable

Don't forget to flip a conversion factor (numerator/denominator) so the unwanted unit is in the position that cancels — a factor used upside-down inverts the whole answer.
PV=k is inverse (P up → V down at constant T), not direct — don't assume all named 'laws' with two variables are directly proportional.

Convert 25°C to Kelvin.

Solve: x + y = 12, x − y = 4. What is x?

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