By trading optimality, completeness, accuracy, and / or precision for speed, a heuristic can quickly produce a solution that is good enough for solving the problem at hand, as opposed to finding all exact solutions in a prohibitively long time.

By using heuristics, time can be reduced when solving problems.

By the first decades of the eleventh century, Chinese ironworkers in the Yellow River valley began to fuel their furnaces with coke, solving their fuel problem in that tree-sparse region.

By forming a partnership with the community through Community Policing Initiatives we will work together to protect the lives and property of the citizens through fair, honest and professional enforcement of the laws, crime prevention and community problem solving.

By solving this problem, online auctions reduce the transaction costs for bidders, increase the number of bidders, and increase the average bid price.

By differentiating both equations one more time and solving them with proper initial conditions

By solving for the fixed point of the system:

By solving the theory in one space and one time dimension ,'t Hooft was able to derive a formula for the masses of mesons.

By using receivers and transmitters tuned to different frequencies, and solving the problems introduced by operation of a receiver immediately next to a transmitter, simultaneous transmission and reception was possible at each end of a radio link, in so-called " full duplex " mode.

By solving the quantum mechanical system of two electrons as harmonic oscillators at some finite distance from one another, being displaced about their respective rest positions and interacting with each others fields, London showed that the energy of this system is given by:

By substituting for y < sup > 2 </ sup > into the equation for triangle BHC and solving for x:

By employing positive group dynamic tools and a rigorous problem solving infrastrucure ( guided by a facilitator ) the team harnesses its multiplicative potential to problem solve.

If housing services ( X4 ) is separated into the components that comprise it ( Z1, Z2, Z3, Z4 ,... Zn ), then the utility function can be rewritten as U = U ( X1, X2, X3 ,( Z1, Z2, Z3, Z4 ,... Zn )... Xn ) By varying the price of housing services ( X4 ) and solving for points of optimal utility, that household's demand schedule for housing services can be constructed.

By his fundamental approach and his inner understanding of human nature, Ohayon succeeds in breaking the ring of silence and solving the murder mystery on his way to the next book.

By solving subproblems one time and reusing these solutions many times, we can drastically reduce the time required.

By solving the riddle ( calculating all the prime numbers below 100 using the Sieve of Eratosthenes and pressing the corresponding buttons backwards ), Roland and his ka-tet are allowed to board Blaine and flee Lud, just as Blaine releases poison gas ( DEP3, a substance similar to what appears as a poison in another of King's books, Eyes of the Dragon.

By solving quests or defeating opponents the player is awarded experience points, which can be used to improve attributes, skills, and weapon proficiencies to further develop the character.

By substituting V < sub > 1 </ sub > to the first equation and solving in respect to V2, we get:

By inducing participants to generate language samples in the context of solving an actual task ( using a computer that they believed actually understood what they were typing ), the variety and complexity of the lexical structures gathered was greatly reduced and simple keyword matching algorithms could be developed to address the actual language collected.

By connecting outward light to inward light, via an interaction point, this equation stands for the whole ' light transport ' — all the movement of light — in a scene.

By direct substitution, the solution to this equation can be readily shown to be the scalar Green's function, which in the spherical coordinate system ( and using the physics time convention ) is:

By a proper choice of coordinate system, the ellipse can be described by the canonical implicit equation

By eliminating ρ and J, using Maxwell's equations, and manipulating using the theorems of vector calculus, this form of the equation can be used to derive the Maxwell stress tensor σ, in turn this can be combined with the Poynting vector S to obtain the electromagnetic stress-energy tensor T used in general relativity.

By interchanging the roles of x and y one obtains the corresponding equation of a parabola with a vertical axis as

By assuming that light actually consisted of discrete energy packets, Einstein wrote an equation for the photoelectric effect that agreed with experimental results.

By using the London equation, one can obtain the dependence of the magnetic field inside the superconductor on the distance to the surface.

By the radiation pressure equation σT < sup > 4 </ sup >/ c ; the sun-facing photon pressure is 3. 61 µPa ( 3. 6 N / km², 2. 08 lbf / mi² ).

By finding the derivative of the equation while assuming that y is a constant, the slope of ƒ at the point is found to be:

By using the definition of an average, this equation states that when the acceleration is constant average velocity times time equals displacement.

By analogy, the magnetic equation is an inductive current involving spin.

By putting the measured period T, and the measured distance between the pivot blades L, into the period equation ( 1 ), g could be calculated very accurately.

By implicit differentiation, one can show that all branches of W satisfy the differential equation

By using the equation of hydrostatic equilibrium, combined with conservation of angular momentum and assuming that the disc is thin, the equations of disk structure may be solved in terms of the parameter.

By Late Antiquity, some Gnostic and Neoplatonist thinkers had expanded this syncretic equation to include Aion, Adonis, Attis, Mithras and other gods of the mystery religions.

By 1928 Dirac deduced his equation from the first successful unified combination of special relativity and quantum mechanics to the electron-the Dirac equation.

By contrast the inverse heat equation, deducing a previous distribution of temperature from final data is not well-posed in that the solution is highly sensitive to changes in the final data.

By the Kelvin – Stokes theorem, this equation can also be written in a " differential form ".

By examining Laplace's equation in spherical coordinates, Thomson and Tait recovered Laplace's spherical harmonics.

By separation of variables, two differential equations result by imposing Laplace's equation:

By combining the beamwidth equation with the gain equation, the relation is:

By quantizing this, we get the non-relativistic Schrödinger equation for a free particle,