Xj is the unknown, AIj is the coefficient, bi is the constant term. Suppose x1=c1, x2=c2... And xn=cn can be substituted into the equation given, then (c1, c2... , cn) is regarded as a solution. If C1, C2,... Cn is not all zero, then (c1, c2... , cn) as a nonzero solution. Assuming that the constant terms are zero, we can call it a homogeneous linear system. In two equations, if the number of unknowns is the same and the solution set is the same, it can be regarded as a cosolution system. This paper uses linear equation to study the influencing factors of ideological and political education in colleges and universities, mainly discusses three problems: first, equations and oil elder sisters, secondly, the number of solving equations, and finally solving equations.
Common solutions are divided into two kinds. One is Clem's rule, which uses inverse matrix to solve linear equations. However, due to the large amount of actual work, it is only used for theoretical analysis. The other is the matrix elimination method, which transforms the augmented matrix of linear equations into a simplified row ladder matrix by using the elementary of rows, which represents the statistics of the linear equations and the original equations of the augmented matrix. In the case that the equations have solutions, the unknown quantities corresponding to the unit column vectors can be regarded as nonfree unknowns, and the other unknowns will be regarded as free unknowns, so the solution of the linear equation can be obtained.
Generally speaking, the firstorder equations of unknown quantities are in the following form:
IIn the above formula, x1, x2,... , xn represents unknown quantity, AIj (1≤ I ≤m, 1≤j≤n) represents equation coefficient, bi (1≤ I ≤m) represents constant term. Coefficients and constant terms belong to all complex numbers or to elements in a field.
In constant terms B1, B2... When bn and bn are all equal to zero, the system of equations is called homogeneous linear equations, and the matrix of m rows and n columns formed by the coefficients is shown as follows:
The above formula represents the coefficient matrix of the system. Add columns with constant terms in A to obtain m row n+1 column matrix, as shown below:
The above equations represent the augmented matrix.
The study of linear equations shows that there are two main results: one is that the necessary conditions for the existence of solutions in linear equations are that the coefficient matrix and the augmented matrix have the same rank; The other is that when the rank of A and is greater than zero, A has A subformula D of order R that is not equal to zero, assuming:
At this point, the equation set and the equation set containing only the first R equations have the same solution, so the first R equations can be rewritten as:
The calculation formula of the above equation is:
In the above formula, Dj (j= 1,2... , r) Replace the JTH column of D with an rorder determinant obtained from the right end column of the system, as shown below:
In the above formula, x1, x2,... , xn represents the rest of the unknowns, xr+1, xr+2... Xn stands for linear free as mass. Under the condition of r Under the condition of r=n, the equations do not contain free unknowns, so the unique solution of the equations should be obtained by combining the above research. Under the condition of m=n=r, the above formula can be regarded as Clem's rule.
Since there are many influencing factors when constructing the research model of ideological and political education teaching effectiveness based on linear equation, multiple linear regression model should be used for deep analysis.
Suppose some dependent variable y is dependent on k independent variables x1, x2... And xN, where the observed values of group N are (YA, X1A, X2a... , xka), then the structural equation of the multiple linear regression model is shown as follows:
In the above formula,
Suppose b0, B1,... Bk stands for
In the above formula, B0 stands for constants, b1, B2... Bk stands for partial regression coefficient.
Combining with the least square method, (I = 0,1,2... The estimated value of k should be bi (I = 0,1,2... , k), and get:
The necessary conditions for calculating extreme values are:
After sorting out the above equations, we can get:
The above equations are called normal equations. Assuming that a vector and matrix are introduced, it can be obtained:
The actual solution is:
Suppose the notation is introduced:
Then the normal equations can also be shown as follows:
Using multiple linear regression model to study the effectiveness of ideological and political education teaching needs to be tested in empirical research. Because the change of observed value of dependent variable is caused by two factors, on the one hand, the value of independent variable is different, on the other hand, it changes under the influence of other random factors, so in order to effectively distinguish, anOVA should be conducted on regression model, the specific formula is as follows:
In the above formula, Sr represents the sum of the squares of deviations, Lyy represents the regression model, U represents the sum of the squares of regression, and Q represents the sum of the remaining squares.
In the analysis of multiple linear regression, the sum of regression squares represents the overall influence of all K variables on the variation of Y, and the actual formula is as follows:
The formula for calculating the sum of the remaining squares is as follows:
At the same time, in multiple linear regression analysis, the degrees of freedom of the sum of squares are different. The degrees of freedom of the sum of squares of regression and U are equal to the number of independent variable K, so the degrees of freedom of the remaining sum of squares are mK1. The calculation formula of F statistic is as follows:
After the statistics are defined, significance test analysis can be performed on the equation model according to the distribution results of F.
In essence, the effectiveness of ideological and political education teaching refers to the attraction and influence of various activities on students during practical teaching. By integrating and referring to the research results proposed by current research scholars, this paper analyzes the effectiveness of ideological and political education from three perspectives: knowledge mastery, teaching effect and overall satisfaction, and puts forward the following three hypotheses: [1]First, students have a positive impact on the effectiveness of ideological and political education teaching; Secondly, students have a positive influence on ideological and political education teaching factors; Finally, students have a positive influence on the teacher factor of ideological and political education teaching. [2.3]
Using modern network technology and thematic questionnaire survey method, this paper conducts research and analysis on 26 universities in many provinces of China, among which school types include double firstclass universities, provincial key universities, ordinary undergraduate, key higher vocational colleges and ordinary higher vocational colleges. Finally, 2,021 valid questionnaires are obtained. [4]
Combined with the above analytical process of multiple linear equations, a structural model is constructed to study the teaching effectiveness of ideological and political education and its influencing factors, as shown in Figure 1 below [5]:
The structural equation model is shown below:
Observe the condition
In the above formula,
Thus, the measurement model of factors influencing the effectiveness of ideological and political education is clarified, and the relationship between latent variables and index variables is shown. It is mainly divided into two aspects: on the one hand, the exogenous latent variable
The above formula meets the condition of
The equation model proposed in this paper and statistical software AMOS 21.0 are used to verify and analyze the model equation proposed in the figure above, and the output results are finally obtained as shown in Figure 2 below:
According to the above analysis, it is found that in the equation model, the practical effect of ideological and political teaching (1), leading factor (2), main factor (1) and teaching factor (3) are latent variables, which do not have negative error variance, and the standard coefficient is controlled between 0.77 and 0.96. At the same time, the model equations were evaluated and analyzed in combination with the evaluation indexes of the three statistics of absolute adaptation, valueadded adaptation and reduced adaptation, and the final results were found to meet the judgment criteria, as shown in Table 1 below [7]:
Fitting results of the equation model
GFI  AGFI  RMSEA  NFI  IFI  REI  PNFI  PGFI  

The initial model  0.897  0.833  0.038  0.942  0.944  0.920  0.685  0.686  2.389 
GFI  AGFI  RMSEA  NFI  IFI  REI  PNFI  PGFI  
The judgment standard  >0.8  >0.8  >0.08  >0.8  >0.8  >0.8  >0.5  >0.5  It's between one and five 
According to the research results of equation model, external factor variables can be regarded as the direct effect, while internal factor variables can be regarded as the indirect effect. According to the output results of Figure 2, the influence of student factors on the effectiveness of ideological and political education is direct, and the actual value can reach 0.16. The influence of teacher factors on the effectiveness of ideological and political education is also direct, and the actual value can reach 0.23. The influence of teaching factors on the effectiveness of ideological and political education is also direct, and the actual value can reach 0.68. Student factor also has direct influence on teacher factor, the actual value is 0.41; Students also have a direct influence on teaching factors. The actual value is 0.15 and the influence of good teachers is also direct, and the actual value can reach 0.60 [8].
From the perspective of indirect effects, if exogenous variables do not directly affect endogenous variables, then intermediary variables are bound to have an impact. Among them, the student factor can have an indirect impact on the effectiveness of ideological and political education. The actual path can be divided into two ways: one is based on the teacher factor, and the actual value is 0.094; The other is based on teaching factors, and the actual value is 0.102. A comprehensive study of these two indirect influence paths shows that the value of students' indirect influence on the effectiveness of ideological and political education is 0.196. The indirect influence path of teacher factors on the effectiveness of ideological and political education should be realized with teaching factors as the core, and the actual value is 0.408.
From the perspective of overall influence, the total influence value of student factors on the effectiveness of ideological and political education can reach 0.356; The total influence of teachers on the effectiveness of ideological and political education can reach 0.683; The total influence of teaching factors on the effectiveness of ideological and political education can reach 0.68. It can be seen that the ranking of the three factors is teaching, teachers and students. The specific results are shown in Table 2 below: [9]
Analysis of influencing factors and effects of ideological and political education's effectiveness
The student factors  Teacher factors  Teaching factors  

Direct effect  Teacher factors  0.41^{***}     
Teaching factors  0.15^{***}      
Practical effect of ideological and political teaching in colleges and universities  0.16^{***}  0.23^{***}  0.68^{***}  
The indirect effect  Teacher factors       
Teaching factors        
Practical effect of ideological and political teaching in colleges and universities  0.196^{***}  0.408^{***}    
The total effect  The student factors  Teacher factors  Teaching factors  
Practical effect of ideological and political teaching in colleges and universities  0.356^{***}  0.638^{***}  0.68^{***} 
Note:
P<0.08,
P<0.05,
P<0.01
According to the above research, the hypothesis verification results are shown in Table 3 below, which proves that student factors have a positive and direct impact on the effectiveness of ideological and political education, and can have a positive and indirect impact on the effectiveness of education by directly affecting teachers and teaching. At the same time, the teacher factor has two influences on the practical effect of ideological and political education, one is direct and the other is indirect. Teaching factors will have a positive and direct impact on teaching effectiveness [10.11].
Research hypothesis and verification results
Research hypothesis  Assume the specific content  The verification results 

H1  The student factor has a positive influence on the practical effect of ideological and political course teaching  through 
H2  Student factors have a positive influence on the teaching factors of ideological and political subject teaching in colleges and universities  through 
H3  Student factors have a positive influence on the teaching factors of ideological and political subject teaching in colleges and universities  through 
H4  The student factor has a positive influence on the practical effect of ideological and political course teaching  through 
H5  Student factors have a positive influence on the teaching factors of ideological and political subject teaching in colleges and universities  through 
H6  The student factor has a positive influence on the practical effect of ideological and political course teaching  through 
Meanwhile, this paper also conducted investigation and research on how to improve the effectiveness of ideological and political education, and the final results are shown in the following table:
Survey results of improving the effectiveness of teaching
move  frequency  Proportion / % 

Enrich teaching content  1255  26.70 
Strengthen classroom management  596  12.68 
Enhance classroom interaction  841  17.89 
Introducing current events  942  20.04 
Change the existing thematic teaching methods  396  8.43 
Adopt a new live broadcasting platform  245  5.21 
Arrange students to teach  99  2.11 
Organize online group discussions  326  6.94 
According to the survey results in the table above, ideological and political education in colleges and universities should be carried out from the following points: First of all, classroom teaching content should be continuously enriched, and scientific adjustment should be made according to the learning needs and abilities of teachers and students in colleges and universities. According to the survey data, 26.7% of students think that ideological and political education needs to be enriched, and the research results of this paper prove that the influence value of teaching content on teaching factors can reach 0.90. Therefore, it is necessary not only to apply diversified teaching cases during classroom teaching, but also to expand the classroom learning path according to the needs of the development of The Times. Secondly, it is necessary to innovate ideological and political education model, get rid of the limitation of traditional teaching concept, and pay attention to the construction of targeted and interesting thematic teaching platform for college teachers and students, so as to generate knowledge and emotional resonance in the interaction and communication. Finally, we should pay attention to the effective optimization of teaching organization and teaching atmosphere, and build a teaching symbiotic mechanism based on the needs of ideological and political education in colleges and universities. Under new era background, platform based on Internet technology constantly optimize the traditional ideological education mode, the correct understanding of mechanism of sharing information resources, information processing, virtual communication content such as the impact on the modern education reform and requirements, in college ideological education guidance during the formation of thinking, interaction, experience of integration of learning mode, build the good learning atmosphere, Pass on excellent ideological and political culture. [12]
To sum up, the linear equation is used to construct a research model for influencing factors of the timeliness of ideological and political education and teaching, so as to systematically understand the guidance status and relevant values of ideological and political education in colleges and universities in recent years. On the basis of clarifying the main influencing factors, the main reasons affecting the timeliness of education and teaching are deeply explored. This article research results show that three factors are students, teachers, teaching has direct and indirect impact on timeliness, so in the education reform should start from building with pertinence and diversity of classroom teaching, active application of advanced means of education and science and technology, only in this way can truly meet the demand of students and education education innovation.
Fitting results of the equation model
GFI  AGFI  RMSEA  NFI  IFI  REI  PNFI  PGFI  

The initial model  0.897  0.833  0.038  0.942  0.944  0.920  0.685  0.686  2.389 
GFI  AGFI  RMSEA  NFI  IFI  REI  PNFI  PGFI  
The judgment standard  >0.8  >0.8  >0.08  >0.8  >0.8  >0.8  >0.5  >0.5  It's between one and five 
Research hypothesis and verification results
Research hypothesis  Assume the specific content  The verification results 

H1  The student factor has a positive influence on the practical effect of ideological and political course teaching  through 
H2  Student factors have a positive influence on the teaching factors of ideological and political subject teaching in colleges and universities  through 
H3  Student factors have a positive influence on the teaching factors of ideological and political subject teaching in colleges and universities  through 
H4  The student factor has a positive influence on the practical effect of ideological and political course teaching  through 
H5  Student factors have a positive influence on the teaching factors of ideological and political subject teaching in colleges and universities  through 
H6  The student factor has a positive influence on the practical effect of ideological and political course teaching  through 
Survey results of improving the effectiveness of teaching
move  frequency  Proportion / % 

Enrich teaching content  1255  26.70 
Strengthen classroom management  596  12.68 
Enhance classroom interaction  841  17.89 
Introducing current events  942  20.04 
Change the existing thematic teaching methods  396  8.43 
Adopt a new live broadcasting platform  245  5.21 
Arrange students to teach  99  2.11 
Organize online group discussions  326  6.94 
Analysis of influencing factors and effects of ideological and political education's effectiveness
The student factors  Teacher factors  Teaching factors  

Direct effect  Teacher factors  0.41 
   
Teaching factors  0.15 
    
Practical effect of ideological and political teaching in colleges and universities  0.16 
0.23 
0.68 

The indirect effect  Teacher factors       
Teaching factors        
Practical effect of ideological and political teaching in colleges and universities  0.196 
0.408 
  
The total effect  The student factors  Teacher factors  Teaching factors  
Practical effect of ideological and political teaching in colleges and universities  0.356 
0.638 
0.68 
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EnglishChinese CrossLanguage Information Retrieval The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization research on prefabricated concrete frame buildings based on the dynamic equation of eccentric structure and horizontaltorsional coupling Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Green building considering image processing technology combined with CFD numerical simulation MCM of Student’s Physical Health Based on Mathematical Cone Sports health quantification method and system implementation based on multiple thermal physiology simulation Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching A mathematical model of PCNN for image fusion with nonsampled contourlet transform Nonlinear Differential Equations in ComputerAided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Design of Tennis Mobile Teaching Assistant System Based on Ordinary Differential Equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with DiamondSquare Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Animation VR scene mosaic modeling based on generalized Laplacian equation Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Crossborder Ecommerce Platform Operation Based on Fractional Differential Equations The influence of accounting computer information processing technology on enterprise internal control under panel data simultaneous equation Research on Stability of Timedelay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Construction of comprehensive evaluation index system of watersaving irrigation project integrating penman Montei the quation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of AirConditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression Analysis of technical statistical indexes of college tennis players under the winlose regression function equation Automatic extraction and discrimination of vocal main melody based on quadratic wave equation Analysis of wireless English multimedia communication based on spatial state model equation Optimization of Linear Algebra Core Function Framework on Multicore Processors Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on shortterm human movement Human gait modelling and tracking based on motion functionalisation The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Nonlinear Differential Equations in Crossborder Ecommerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model Fractional Differential Equations in Electronic Information Models BIM Engineering Management Oriented to Curve Equation Model Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Design of language assisted learning model and online learning system under the background of artificial intelligence Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering Action of Aut( G ) on the set of maximal subgroups ofp groupsThe internal mechanism of corporate social responsibility fulfillment affecting debt risk in China: analysis of intermediary transmission effect based on degree of debt concentration and product market competitive advantage Study on transmission characteristics in three kinds of deformed finlines based on edgebased finite element method Asymptotic stability problem of predator–prey system with linear diffusion Research on loyalty prediction of ecommerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Research on threat assessment problems of island air defence system based on the leaderfollower model Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on ecommerce platform Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long shortterm memory networks Evaluation of ecosystem health in Futian mangrove wetland based on the PSRAHP model A study of local smoothnessinformed convolutional neural network models for image inpainting Towards more efficient control of the ironmaking blast furnace: modelling gaseous reduction of iron ores in H_{2}N_{2} atmosphere Algorithm of overfitting avoidance in CNN based on maximum pooled and weight decay Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Crossborder Ecommerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model ChineseEnglish Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model