Specific method
Substitution elimination method (commonly used, see 2. 1).
Addition, subtraction and elimination (commonly used, see 2.2 for methods)
Sequential elimination method (often used in computers, the method is as follows)
Sequential exclusion method
Set a set of binary linear equations,
If so, the formula (3) is obtained:
In the formula (3)
Then you can find the root formula:
The above process is called "sequential elimination method", and the principle of solving multivariate equations is the same.
This method is widely used in computers because there is only the operation between numbers without the operation of the whole formula. When solving a mathematical problem, we regard a formula as a whole and replace it with a variable, thus simplifying the problem. This is called substitution. The essence of substitution is transformation, the key is to construct elements and set elements, and the theoretical basis is equivalent substitution. The purpose is to change the research object, move the problem to the knowledge background of the new object, standardize non-standard problems, simplify complex problems and become easy to deal with.
Substitution method is also called auxiliary element method and variable substitution method. By introducing new variables, scattered conditions can be linked, implicit conditions can be revealed, or conditions can be linked with conclusions. Or turn it into a familiar form to simplify complicated calculation and derivation.
It can transform high order into low order, fraction into algebraic expression, irrational expression into rational expression, transcendental expression into algebraic expression, and is widely used in the study of equations, inequalities, functions, sequences, triangles and other issues.
take for example
(x+y)/2-(x-y)/3=6①
3(x+y)=4(x-y)②
Solution: let x+y be a and X-Y be b.
So, the original equation becomes
a/2-b/3=6③
3a-4b=0 ④
Solution:
a=24
b= 18
Therefore:
x+y=24
x-y= 18
The solution of the equation is:
x= 2 1
y= 3 x:y= 1:4 ①
5x+6y=29 ②
Solution: let x = t and y = 4t, then equation ② can be written as: 5t+6×4t=29→29t=29→t= 1, so x = 1, y = 4.
The derivation process of binary linear equations;
There is only one unknown number y in the final formula, and the value of y (y=? ), and then substitute a1x+b1y = k1; Look for X. Example:
Y = (2-3/4× 0)/(1-3/4 *×) = 2/(-1/2) =-43x-4 = 2 or 4x-8=0 x=2 Derive a simple equation:
Equation = 0; Unknown number 0; 1 There is a binary linear equation system ① today.
Let matrix A=, vector sum, defined by the product of matrix and vector, and then compare the equations, we can know that there is the following relationship:
②
We call ② the matrix form of the system of equations ①.
Matrix a can be regarded as a linear transformation p, that is, the vector is transformed according to the linear transformation p to get the vector. Therefore, the process of solving the equation can be regarded as finding a vector, which is obtained by linear transformation P. Because this is the process of finding a vector, it can also be called solving a vector.
Intuitively understand the above sentence. For example, if a vector A rotates 30 counterclockwise to get a new vector B, then after B rotates 30 clockwise, A can definitely be obtained. For example, if you expand the vertical and horizontal coordinates of a vector A by n times to get a vector B, then if you reduce the vertical and horizontal coordinates of B by n times, you will definitely get A ... So, given the relationship between B and linear transformation, A is the solution of the equation.
The linear transformation of matrix A and its inverse matrix is reciprocal, so the process of finding the vector is equivalent to finding the inverse matrix of the matrix. According to the properties of matrices, the necessary and sufficient condition for a matrix to have an inverse matrix is a second-order determinant. Therefore, the necessary and sufficient condition for the equations to have a solution is ad-bc≠0.
According to the solution of the inverse matrix, the inverse matrix is
In other words, the solution of the equation is
This method can also be used as the root formula of binary linear equations. (the premise is that ad-bc≠0! )
example
Solving binary linear equations by solution vector method
In this problem, a=3, b= 1, c=4, d=2, e=2, f=0, ad-bc=3*2- 1*4=2≠0.
∴ Equation has a solution, and this solution is
x =(de-BF)/(ad-BC)=(2 * 2- 1 * 0)/2 = 2
y =(af-ce)/(ad-BC)=(3 * 0-4 * 2)/2 =-4