Solution ideas:
Let's start by constructing a complex system of ternary quadratic equations, for example:
1)x^22y-z=7
2)y^2-xz3z=10
3)z^2xy-4y=13
This system of equations contains three unknowns x, y, and z, and each equation is of quadratic form. Our goal is to find the real solution to this system of equations.
The steps to solve the problem are as follows:
Step 1: Elimination and substitution.
Try to convert a system of ternary equations into a system of binary equations or even a unitary equation by the elimination method or substitution method. For example, we can start by solving z:
Z=x 22y-7 (Equation 1).
This expression is then substituted into the second and third equations to eliminate the z-variable, resulting in two quadratic equations for x and y.
Step 2: Convert into a system of two-dimensional quadratic equations.
After substitution, we get the new equation:
1)y^2-x(x^22y-7)3(x^22y-7)=10
2)(x^22y-7)^2x(y^2-x(x^22y-7)3(x^22y-7))-4y=13
These two equations are simplified and organized into a standard two-element quadratic equation form.
Step 3: Solve the system of two-dimensional quadratic equations.
Using Vedica's theorem or the matching method, factorization and other methods to solve this two-element quadratic equation system, the values of x and y are obtained.
Step 4: Solve z
The values of z can be obtained by substituting the values of x and y into the expression about z derived from the first equation (Equation 1).
It should be noted that in the actual solution process, there may be no solution, solution or multiple solutions, which need to be judged according to the specific situation. Due to space limitations, the specific calculation process is not given here, but the basic strategies and methods for solving such complex ternary quadratic equations have been elaborated in the above steps. In real situations, tedious calculations may be required with the help of mathematical software or calculators.
(x^22y-8)^3x(y^1-x(x^12y-5)3(x^22y-9))-4y=12；