RESOLUTION, in algebra, or algebrai cal resolution, is of two kinds ; the one practised in numerical problems, the other in geometrical ones.
In resolving a numerical problem alge braically, the method is this : First, the given quantities are distinguished from those that are sought ; and the former denoted by the initial letters of the alpha bet, but the latter by the last letters. 2. Then as many equations are formed as there are unknown quantities. If that cannot be done from the proposition or data, the problem is indeterminate : and certain arbitrary assumptions must be made to supply the defect, and which can satisfy the question. When the equations are not in the problem itself, they are to be found by particular theo rems concerning equations, ratios, pro portions, &c. Since, in an equation, the known and unknown quantities are mix ed together, they must be separated in such a manner that the unknown remain alone on one side, and the known ones on the other. This reduction, or separation, is made by addition, subtrac tion, multiplication, division, extraction of roots, and raising of powers ; resolving every kind of combination of the quanti ties by their counter or reverse ones, and performing the same operation on all the quantities, or terms on both sides of the equation, that the equality may still be preserved.
To resolve a geometrical problem alge braically. The same sort of operations are to be performed as in the former arti cle : besides several others, that depend upon the nature of the diagram, and geo metrical properties. As, 1. The thing required or proposed, must be supposed done, the diagram being drawn or con structed in all its parts, both known and unknown. 2. We must then examine the geometrical relations which the lines of the figure have among themselves, without regarding whether they are known or unknown, to find what equa tions arise from those relations for finding the unknown quantities. 3. It is often necessary to form similar triangles and rectangles, sometimes by producing of lines, or drawing parallels and perpendi culars, and forming equal angles, &c. ; till equations can be formed from them, including both the known and unknown quantities.