# Solving Equations

• The purpose of solving a single-variable equation is to find the zero or root of the function , at which . Graphically, the root of a function can be found in the x-y plane as the intersection of the curve with the horizontal axis for , if they do intersect, i.e., if the root exists. A function may have zero root such as , a unique root such as for , or multiple roots such as for .

• The solution of a 2-variable equation system composed of and are the points in the x-y plane at which both functions are zero. Graphically, a function can be represented as a surface in a 3-D space with the third (vertical) dimension for the value of the function at the point at . The roots of this function, if exist, are the intersection of the surface and the x-y plane, a curve on the x-y plane. On one side of the curve , while on the other side . If has no solution, then the intersection does not exist. The roots for both and are the points on the intersections of the two curves representing the individual roots of the two function, if they do intersect. Again there may be zero, one, or multiple such points. • To solve N simultaneous N-variable equations , we need to find all points in an N-D vector space at which all equations are zero. This problem can be viewed in an dimensional space, in which the value of each function is represented in the (N+1)th dimension as a function of the N-D points in the space formed by the remaining dimensions, i.e., a hyper-surface in the N+1 dimensional space. Moreover, the roots of the equation is an N-D hyper-surface, which is the intersection of the hyper-surface in the (N+1)th dimension with the N-D space representing the variables. The solutions of all the equation system are therefore the intersections of all such hyper-surfaces.

Example: Consider a simultaneous equation system: The first function is a parabolic cone centrally symmetric to the vertical axis, and its roots form a circle on the x-y plane centered at the origin with radius ; the second function is a plane through the origin, and its roots form a straight line on the x-y plane. The roots of the equation system are where the two curves intersect:

• If , there are two roots and ;
• If , there is only one root ;
• If , the two curves do not intersect, i.e., there are no roots.

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