Solving Equations

• The purpose of solving a single-variable equation $f(x)=0$ is to find the zero or root $x^*$ of the function $f(x)$, at which $f(x^*)=0$. Graphically, the root of a function $y=f(x)$ can be found in the x-y plane as the intersection of the curve $y=f(x)$ with the horizontal axis for $y=0$, if they do intersect, i.e., if the root exists. A function may have zero root such as $f(x)=x^2+1>0$, a unique root such as $x=0$ for $f(x)=x^3$, or multiple roots such as $x=\pm 1$ for $f(x)=x^2-1$.

• The solution of a 2-variable equation system composed of $f_1(x,y)=0$ and $f_2(x,y)=0$ are the points $(x,y)$ in the x-y plane at which both functions are zero. Graphically, a function $f(x,y)$ can be represented as a surface in a 3-D space with the third (vertical) dimension for the value of the function $z=f(x,y)$ at the point at $(x,\,y)$. 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 $f(x,\,y)>0$, while on the other side $f(x,\,y)<0$. If $f(x,y)=0$ has no solution, then the intersection does not exist. The roots for both $f_1(x,y)$ and $f_2(x,y)$ 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 $f_i(x_1,\cdots,x_N)=0,\;\;(i=1,\cdots,N)$, we need to find all points $(x_1,\cdots,x_N)$ in an N-D vector space at which all $N$ equations are zero. This problem can be viewed in an $N+1$ dimensional space, in which the value of each function $f_i(x_1,\cdots,x_N)$ is represented in the (N+1)th dimension as a function of the N-D points $(x_1,\cdots,x_N)$ 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 $f_i(x_1,\cdots,x_N)=0$ 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 $N$ variables. The solutions of all the equation system are therefore the intersections of all $N$ such hyper-surfaces.

Example: Consider a simultaneous equation system:

$$\left\{\begin{array}{ll} f_1(x,y)=x^2+y^2-2=0\\ f_2(x,y)=x+y-C=0\end{array}\right.$$

The first function $f_1(x,y)$ is a parabolic cone centrally symmetric to the vertical axis, and its roots form a circle $x^2+y^2=2$ on the x-y plane centered at the origin $(0,\,0)$ with radius $1$; the second function $f_2(x,y)$ is a plane through the origin, and its roots form a straight line $y=C-x$ on the x-y plane. The roots of the equation system are where the two curves intersect:

• If $C=0$, there are two roots $(1,\,-1)$ and $(-1,\,1)$;
• If $C=2$, there is only one root $(1,\,1)$;
• If $C=3$, the two curves do not intersect, i.e., there are no roots.