The solution of binary linear equations is essentially to transform binary linear equations into univariate linear equations by using the idea of mathematical transformation. The specific transformation method is to eliminate the "two unknowns" in the binary linear equations by using "substitution elimination method" or "addition and subtraction elimination method", so as to obtain the linear equations, thus realizing the transformation from "unknown" to "known" and then solving them. There are rich mathematical thinking methods here, which I gradually permeate to students in teaching. Here is an example:
First, flexible use of substitution method, clever evaluation:
The substitution method is to solve binary linear equations. One equation in the equations is transformed into a mathematical formula containing an unknown number to represent another unknown number, and then it is substituted into another equation, so as to eliminate an unknown number, get a linear equation and then solve it. With the help of this method, the conventional fixed value problem can be solved.
Example 1. If 5x-6y=0 and xy≠0, the value of is equal to.
Solution. Substitute 5x=6y into the solution from 5x-6y=0: 5x=6y.
Reflection: This problem can be easily solved by method of substitution.
Variant exercise: If 2x-3y=0 and xy≠0, the value of is equal to.
Example 2. If 4x+3y+5=0, the value of 3 (8y-x)-5 (x+6y-2) is equal to _ _ _ _ _ _ _;
Analysis: It is easy to know through the exam. You can simplify 3 (8y-x)-5 (x+6y-2) first.
-8x-6y+ 10, and then it is easy to find its value by replacing it in whole or in part.
Solution: ∫4x+3y+5 = 0,
∴4x+3y=-5
3(8y-x)-5(x+6y-2)
= 24 y-3x-5x-30y+ 10
=-8x-6y+ 10
=-2(4x+3y)+ 10
=-2×(-5)+ 10
=20
Reflection: This question can also be changed to 4x+3y+5=0 to get x =-.
Second, the clever use of addition and subtraction, rapid evaluation:
Addition and subtraction is to change the coefficient of an unknown in an equation into the same or opposite number, and then add or subtract it with two equations, that is, subtract when the coefficient of an unknown becomes the same; When the coefficient of an unknown becomes a reciprocal, it is added, thus eliminating an unknown, obtaining a linear equation and solving it. In addition, the rational use of addition and subtraction in evaluation questions can get twice the result with half the effort.
Example 3. If 2x+3y= 16 and 3x+2y= 19, then.
Analysis: If 2x+3y= 16 and 3x+2y= 19 are directly combined to solve the equations, the amount of calculation is very large, and it is easy to make mistakes when substituting them for solution evaluation. If you carefully analyze the numerical formula, you can consider adding and subtracting to get the values of x+y and x-y quickly, and this problem will be solved.
Solution: from the meaning of the question:
From 1+2: 5x+5y=35.
x+y=5
From 2- 1: X-Y = 3.
therefore
x=4，y= 1
Note: If this problem is regarded as a binary linear equation system about X and Y, it will be very complicated to find the values of X and Y first and then substitute them into the calculation. If the basic thinking method of "addition and subtraction" is skillfully used, it will receive miraculous effects. Third, change the "unknown" into "known" and infiltrate the transformation.
Solutions of linear equations; Calculation of matrix eigenvalues and eigenvectors; Nonlinear equation and iterative solution of nonlinear equation; Interpolation and approximation; Numerical integration; Numerical solution of initial value problem of ordinary differential equation and difference decomposition of partial differential equation. The content is rich and systematic, and the depth and breadth are suitable for the training requirements of engineering master students. The language of this book is concise and fluent, and the numerical examples and exercises are very rich.