A system of equations requires one choice of unknowns to satisfy several equations at once. Its solution set is the intersection of the solution sets of the individual equations. For example, $x^2+y^2=1$ and $y=(x-1{)}^2$ ask for intersections of a unit circle and a parabola. A linear system is the special case in which every equation is linear in the unknowns.

An implicit equation specifies a relation between variables without isolating a designated output as an explicit function of the inputs. For example, $x^2+y^2=1$ specifies a circle. On $-1<x<1$, it determines the two real branches $y=\pm \sqrt{1-x^2}$, each an implicit function when one branch is selected. The implicit function theorem gives local conditions for a unique differentiable branch; implicit differentiation computes its derivative without first solving explicitly.

Excitation energy is an energy level's energy above a specified ground state. Emitted photon energies measure differences between levels, with a small recoil correction when the emitter has finite mass.

A nuclear energy level is an energy eigenvalue of an atomic nucleus. Differences between levels give nuclear excitation energies and the approximate gamma ray energies of transitions.

An energy barrier is an increase in energy that must be overcome along an allowed route between states. Its value depends on which paths or single-step moves are allowed; specifying only the energies of the endpoints does not determine the barrier.

Binding energy is the energy required to separate a bound state into specified free constituents. For an electron bound in a chemical atom, neglecting this cost is an assumption of the free-electron Compton scattering model.

The Q value of a reaction or particle decay is the initial total rest energy minus the final total rest energy. Positive $Q$ releases kinetic energy; a negative $Q$ requires energy supplied by incident motion or another source. Atomic and bare nuclear mass conventions must be distinguished when counting electrons in beta decay.

A microsome is a closed lipid bilayer compartment produced during disruption of a biological tissue; rough endoplasmic reticulum-derived microsomes provide a sealed compartment for protein targeting experiments.

Secretory protein routing connects entry into the endoplasmic reticulum, protein folding and sorting, transfer to the Golgi apparatus, and delivery onward. In the conventional soluble-cargo route, transport vesicles carry lumenal contents between compartments; exocytosis releases cargo outside the cell membrane. Resident endoplasmic reticulum proteins can be retained or retrieved, and defective cargo can be held back. Many cytosol proteins never enter this route.^[ref]

The useful geometric rule is preservation of membrane sides: during budding and fusion, a lumen-facing region of a membrane protein stays lumen-facing, then faces the exterior after exocytosis; its cytosol-facing region stays cytosolic.^[ref]

Work through the schematic using two distinguishable labels A and B on a stipulated membrane protein. Start with A facing the ER lumen and B facing the cytosol. At each budding or fusion step, carry the surrounding membrane together with the labels; do not turn the protein over. The resulting pair is A facing the Golgi apparatus interior and B facing the cytosol, then A facing a transport vesicle interior and B facing the cytosol, and finally A facing the exterior and B facing the cytosol. This is a stepwise argument about the depicted operations, not a claim that the compartments have identical chemical contents. Likewise, once soluble cargo is inside the ER lumen, this schematic reaches the exterior with zero further crossings through a lipid bilayer: enclosed spaces join when membranes fuse. Delivery therefore need not mean repeatedly pushing a folded protein through a membrane.