Representing, Solving, and Using Algebraic Equations

1989 ◽  
Vol 82 (8) ◽  
pp. 608-612
Author(s):  
Joe Dan Austin ◽  
H. J. Vollrath
Keyword(s):  
Linear Equations ◽  
Algebraic Equations ◽  
Beginning Algebra ◽  
Physical Problems ◽  
Solving Equations ◽  
Physical Objects

Students of beginning algebra are quickly expected to solve linear equations. The solution procedures are generally abstract, involving the manipulation of numbers and algebraic symbols. Many students, even after completing a year of algebra, do not understand variables, equations, and solving equations (cf. Carpenter et al. [1982]). One way to help students learn to solve equations is to use physical objects, diagrams, and then symbols to represent equations. (Bruner [1964, 1967] calls such representations enactive (concrete), iconic (pictorial), and symbolic.) Although solving equations symbolically is essential, many students can benefit from working with physical problems that can also be symbolized mathematically. This article describes one way for students to learn to solve certain linear equations using pan balances, diagrams, and then symbols.

scholarly journals INDOOR AIR TEMPERATURE DISTRIBUTION – AN ALTERNATIVE APPROACH TO BUILDING SIMULATION

2003 ◽  
Vol 2 (1) ◽  
Author(s):  
A. T. Franco ◽  
C. O. R. Negrão
Keyword(s):  
Temperature Distribution ◽  
Air Temperature ◽  
Indoor Air ◽  
Linear Equations ◽  
Three Dimensional ◽  
Momentum Conservation ◽  
Conservation Equation ◽  
Algebraic Equations ◽  
Conservation Of Energy ◽  
Energy Conservation Equation

The current paper presents a model to predict indoor air temperature distribution. The approach is based on the energy conservation equation which is written for a certain number of finite volumes within the flow domain. The magnitude of the flow is estimated from a scale analysis of the momentum conservation equation. Discretized two or three-dimensional domains provide a set of algebraic equations. The resulting set of non-linear equations is iteratively solved using the line-by-line Thomas Algorithm. As long as the only equation to be solved is the conservation of energy and its coefficients are not strongly dependent on the temperature field, the solution is considerably fast. Therefore, the application of such model to a whole building system is quite reasonable. Two case studies involving buoyancy driven flows were carried out and comparisons with CFD solutions were performed. The results are quite promising for cases involving relatively strong couplings between heat and airflow.

A Diagonally Dominant Solution for the Cylinder End Problem

1981 ◽  
Vol 48 (4) ◽  
pp. 876-880 ◽  
Author(s):  
T. D. Gerhardt ◽  
Shun Cheng
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Infinite Cylinder ◽  
Algebraic Equations ◽  
Precise Calculation ◽  
Linear Algebraic Equations ◽  
Present Solution ◽  
Diagonally Dominant ◽  
Expansion Coefficients

An improved elasticity solution for the cylinder problem with axisymmetric torsionless end loading is presented. Consideration is given to the specification of arbitrary stresses on the end of a semi-infinite cylinder with a stress-free lateral surface. As is known from the literature, the solution to this problem is obtained in the form of a nonorthogonal eigenfunction expansion. Previous solutions have utilized functions biorthogonal to the eigenfunctions to generate an infinite system of linear algebraic equations for determination of the unknown expansion coefficients. However, this system of linear equations has matrices which are not diagonally dominant. Consequently, numerical instability of the calculated eigenfunction coefficients is observed when the number of equations kept before truncation is varied. This instability has an adverse effect on the convergence of the calculated end stresses. In the current paper, a new Galerkin formulation is presented which makes this system of equations diagonally dominant. This results in the precise calculation of the eigenfunction coefficients, regardless of how many equations are kept before truncation. By consideration of a numerical example, the present solution is shown to yield an accurate calculation of cylinder stresses and displacements.

scholarly journals On the Solubility of Linear Algebraic Equations (Contd.)

1913 ◽  
Vol 12 ◽  
pp. 137-138
Author(s):  
John Dougall
Keyword(s):  
Linear Equations ◽  
Homogeneous System ◽  
Algebraic Equations ◽  
Similar Reasoning ◽  
Null Solution ◽  
Linear Algebraic Equations ◽  
The Given ◽  
Homogeneous Linear

A system of n non-homogeneous linear equations in n variables has one and only one solution if the homogeneous system obtained from the given system by putting all the constant terms equal to zero has no solution except the null solution.This may be proved independently by similar reasoning to that given for Theorem I., or it may be deduced from that theorem. We follow the latter method.

scholarly journals A New Operational Matrices-Based Spectral Method for Multi-Order Fractional Problems

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1471 ◽  
Author(s):  
M. Hamid ◽  
Oi Mean Foong ◽  
Muhammad Usman ◽  
Ilyas Khan ◽  
Wei Wang
Keyword(s):  
Applied Sciences ◽  
Collocation Method ◽  
Computational Method ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Physical Problems ◽  
Nonlinear Algebraic Equations ◽  
New Type ◽  
Error Bound Analysis ◽  
Bound Analysis

The operational matrices-based computational algorithms are the promising tools to tackle the problems of non-integer derivatives and gained a substantial devotion among the scientific community. Here, an accurate and efficient computational scheme based on another new type of polynomial with the help of collocation method (CM) is presented for different nonlinear multi-order fractional differentials (NMOFDEs) and Bagley–Torvik (BT) equations. The methods are proposed utilizing some new operational matrices of derivatives using Chelyshkov polynomials (CPs) through Caputo’s sense. Two different ways are adopted to construct the approximated (AOM) and exact (EOM) operational matrices of derivatives for integer and non-integer orders and used to propose an algorithm. The understudy problems have been transformed to an equivalent nonlinear algebraic equations system and solved by means of collocation method. The proposed computational method is authenticated through convergence and error-bound analysis. The exactness and effectiveness of said method are shown on some fractional order physical problems. The attained outcomes are endorsing that the recommended method is really accurate, reliable and efficient and could be used as suitable tool to attain the solutions for a variety of the non-integer order differential equations arising in applied sciences.

scholarly journals On the economical solution method for a system of linear algebraic equations

2004 ◽  
Vol 2004 (4) ◽  
pp. 377-410 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Vadim A. Krysko ◽  
Anton V. Krysko
Keyword(s):  
Differential Equations ◽  
Boundary Value ◽  
Linear Equations ◽  
Three Dimensional ◽  
Boundary Element Methods ◽  
Minimal Amount ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Point Of Intersection ◽  
Stationary Heat Transfer

The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped inℝ3is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error ofO(hx12+hx22). The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.

Symbolic Linearized Equations for Nonholonomic Multibody Systems With Closed-Loop Kinematics

2018 ◽  
Author(s):  
Márton Kuslits ◽  
Dieter Bestle
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Closed Loop ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Algebraic Equations ◽  
Computationally Efficient ◽  
Linearized Equations ◽  
Nonlinear Equations Of Motion ◽  
Loop Constraints

Multibody systems and associated equations of motion may be distinguished in many ways: holonomic and nonholonomic, linear and nonlinear, tree-structured and closed-loop kinematics, symbolic and numeric equations of motion. The present paper deals with a symbolic derivation of nonlinear equations of motion for nonholonomic multibody systems with closed-loop kinematics, where any generalized coordinates and velocities may be used for describing their kinematics. Loop constraints are taken into account by algebraic equations and Lagrange multipliers. The paper then focuses on the derivation of the corresponding linear equations of motion by eliminating the Lagrange multipliers and applying a computationally efficient symbolic linearization procedure. As demonstration example, a vehicle model with differential steering is used where validity of the approach is shown by comparing the behavior of the linearized equations with their nonlinear counterpart via simulations.

Cognitive Models Underlying Students' Formulation of Simple Linear Equations

1993 ◽  
Vol 24 (3) ◽  
pp. 217-232 ◽  
Author(s):  
Mollie MacGregor ◽  
Kaye Stacey
Keyword(s):  
Natural Language ◽  
Linear Equations ◽  
Cognitive Model ◽  
Multiple Choice ◽  
Response Format ◽  
Algebraic Equations ◽  
Free Response ◽  
Multiple Choice Item ◽  
Reversal Error ◽  
Natural Language Statement

Data are presented to show that errors in formulating algebraic equations are not primarily due to syntactic translation, as has been assumed in the literature. Furthermore, it is shown that the reversal error is common even when none of the previously published causes of the error is applicable. A new explanation is required and is proposed in this paper. An examination of students' errors leads us to suggest that students generally construct from the natural language statement a cognitive model of compared unequal quantities. They formulate equations by trying to represent the model directly or by drawing information from it. This hypothesis is supported by research on the comprehension of relationships by linguists, pyscholinguists and psychologists. Data were collected from 281 students in grade 9 in free response format and from 1048 students in grades 8, 9, and 10 who completed a multiple-choice item.

Solving Equations: Exploring Instructional Exchanges as Lenses to Understand Teaching and Its Resistance to Reform

2019 ◽  
Vol 50 (1) ◽  
pp. 51-83 ◽  
Author(s):  
Orly Buchbinder ◽  
Daniel I. Chazan ◽  
Michelle Capozzoli
Keyword(s):  
Teacher Beliefs ◽  
Mathematics Teaching ◽  
Explanatory Variable ◽  
Linear Equations ◽  
Classroom Interaction ◽  
Media Representations ◽  
Theoretical Construct ◽  
Mathematics Classrooms ◽  
Rich Media ◽  
Solving Equations

Many research studies have sought to explain why NCTM's vision for mathematics classrooms has not had greater impact on everyday instruction, with teacher beliefs often identified as an explanatory variable. Using instructional exchanges as a theoretical construct, this study explores the influence of teachers' institutional positions on the solving of equations in algebra classrooms. The experimental design uses surveys with embedded rich-media representations of classroom interaction to surface how teachers appraise correct solutions to linear equations where some solutions follow suggested textbook procedures for solving linear equations and others do not. This paper illustrates the feasibility of studying teaching with rich-media surveys and suggests new ways to support changes in everyday mathematics teaching.

Two Car-Buying Strategies: An Old Problem Revisited with a New Analysis

1996 ◽  
Vol 89 (3) ◽  
pp. 196-199
Author(s):  
Tapan Sen ◽  
Azar D. Raiszadeh ◽  
Farhad M. E. Raiszadeh
Keyword(s):  
Secondary School ◽  
Finite Length ◽  
Applied Mathematics ◽  
Secondary School Student ◽  
Beginning Algebra ◽  
Solving Equations ◽  
Spreadsheet Analysis

This article is an extension of one published in this journal by Bland and Givan (1983). The objective of that piece was “to present a problem in applied mathematics that can be used as a study project by a secondary school student.” The problem addressed applications of sequences and series. It touched on a variety of topics, such as beginning algebra, the theory of interest, and the theory of sums of series of finite length. This article expands the scope of the previous work and demonstrates the application of spreadsheet analysis in solving equations beyond quadratics.

Using, Seeing, Feeling, and Doing Absolute Value for Deeper Understanding

2008 ◽  
Vol 14 (4) ◽  
pp. 234-240
Author(s):  
Gregorio A. Ponce
Keyword(s):  
Linear Equations ◽  
Absolute Value ◽  
Beginning Algebra ◽  
False Position ◽  
False Assumption

Several activities that are based on the ancient method of false position, also called false assumption, are presented in this article as a way to motivate students to find the solution of literal equations in beginning algebra. With a historical connection, the activities described engage students in solving linear equations in a novel way. Extension exercise included.

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