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3 Solving Linear Equations

Notes

This folder Solving Linear Equations offers lesson ideas for teaching in high school or college. Above average students may be able to master all four steps and substeps below quickly.

1. Subfolder Stick Diagrams gives a concrete and visual context for many of the rules or patterns for solving linear equations ax+b = cx+ d. The context may develop equation solving skills and confidence. The idea is to build up student confidence in problem solving before presenting any formal algebraic statement of the rule and patterns for solving equations.

   Here the solution of Linear Equations ax+b = cx+ d with stick diagrams employs addition, subtraction, multiplication and division operations on pairs of sticks or line segments. A three column format allows solutions steps to be done and recorded in an observable manner. Solutions here are accompanied by checks. Showing how to solve and how to check solutions provides a path to follow.

   Before meeting stick diagrams, students should be familar with formulas for perimeters, areas and even volumes in which letters name or denote lengths to be given, lengths often visible in diagrams. Likewise, stick diagram, letters are also used to indicate lengths of visible line segments, but here instead of being given, the lengths are to be found. The stick diagram methods keeps the length visible while performing operations on a pair of stick until the unknown length is found. There-in lies the concrete and visual context for this approach.

   In this stick diagram approach, students will see the domino effects of errors, learn to do and record steps in an observable and verifiable or correctable manner, learn to check results, learn that when a check fails thatthe error or errors may be found between the start of the solution and the end of the check, and see the algebra column, the corresponding algebraic ways for solving equations ax+b=cx+d.

   The twin objectives of this approach are (i) to introduce more general, stick free, algebraic approach; and (ii) to develop or reinforce fraction skills and sense. Once (i) and (ii) have been mastered, students should be encouraged to solve linear equations without the use of stick diagrams. Teachers: Do not be surprised if some students cannot make the transition immediately. Keep on trying, or show how to do the algebraic approach directly.

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2. Subfolder Algebraic Solutions for One Unknown in lessons 2 to 4 offers examples of solving linear equations with whole number and fractional coefficients. Solution checks are usually includedd - emphasize when a check fails that the error or errors may be found between the start of the solution and the end of the check.

   Lesson 4 reproduces the first part of Volume 2, Three Skills for Algebra, Chapter 15. In this lesson, equations of the form ax +c = d are repeated solved to lead students to see (we hope)the derivation of the solution formula x = (d-c)/a as an algebraic shorthand description of the numerical solutions. An algebraic check is included. Then solution formula is employed to solve many numerical examples. Lesson 4 aims to provide a step by step path into the algebraic way of reasoning with letters and symbols, with the understanding that they are place holders for numbers. Following in the footsteps of Lesson 4, Lesson 5 derives algebraic solution of the more general equation ax + b = cx +d.

   Before solving linears numerically, students should be familar formulas for perimeters, areas and volumes. These formulas describes many possible calculation, all at once, not all of which have to be done. After lessons 4 and/or 5, students may see the further power of algebra to solve many like or similar problems at once.

   In numerical examples or exercises involving the solution of linear equations in one unknown with integral and fractional coefficients, students should see again the domino effects of errors, should do and record steps in an observable and verifiable or correctable manner, should check results and be aware that when a check fails thatthe error or errors may be found between the start of the solution and the end of the check. Solutions should further develop and reinforce calculation skills with fractions, but only after student success in solving linear equations with integral coefficients that have integral solutions, non-negative in the first instance.

   N.B. The literal or algebraic solutions of equations is introduced further in a small example in Chapter 10 and in many examples in Chapter 14 of site Volume 2, Three Skills for Algebra. Chapter 14 in particular emphasizes the forward and backward use of a formula - the compound interest formula, while contrast arithmetic (a.k.a numerical) and algebraic (a.k.a literal) solutions. The careful forward and backward use of rules and formulas, proportionality relations included, represents a unifying thread for mathematical and scientific subjects in secondary and college.

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3. In the subfolder Easy systems in 2 or more unknowns, the easy systems are provided by (i) groups of equations in essentially one unknown and systems of equations that are triangular - or become so after a changing the order of the equations. Students may be surprised in being informed that a single letter has or should have one and only one value in the "simultaneous" equations forming a systems. That is contrary to their experience in solving linear equations - one isolated equation at a time.

   In triangular systems, one equation by itself gives the value of one unknown. Then another equation by itself gives the value of a second unknown. Whence the unknowns can be made known one at a time, one after another. Recognizing triangular system and how to solve also the set the stage for transforming general systems into essentially triangular form.

   In systems in essentially one unknown, all unknowns are expressed in terms of one - the key or essential unknown. By one or more substitutions, a single equation in the key or essential unknown results. Solution of the latter equation then gives the value of the key or essential unknown. The derivation of the latter equation forces an operational if not formal command of associative laws for multiplication and/or distributive laws for multiplication over addition. Once the value of the key or essential unknown is found, the values of the other unknowns can be obtained.

   The substitution operations which turn one of the equations into a single equation in one unknown also provide another partial model for solution of linear systems in two or more unknowns. Altogether, the solution of systems in essentially one unknown helps build arithmetic and algebraic skills and confidence.

   Many of the harder word problems in junior high school mathematics may be cast as systems of equation in essentially one unknown, and then solved algebraically. The discussion here of systems of in essentially one unknown makes that easier. Alternatively, apart from the mastery of such systems, students can endeavour to express all numbers and quantities in a given word problem in terms of one unknown in a way that leads to a single equation in the latter. On page 77 in the book

   Problem Solving Through Recreational Mathematics, Averbach and Chern, year 2000 edition,

   solves an simple problem in three unknown ages with both approaches, in a side-by-side two column format, given for the sake of comparison. Opinion: The mechanical formulation of junior high school word problem as a system of linear equations, the form of which identifies the key or essential unknown, provide the simplest approach to such problems.

   The treatment here of both kinds is accompanied by instruction on how to check solutions. Here again if the check fails, the error or errors lies between the first line of the solution and the last line of the check.

   Gifted students can be invited to solve triangular and/or essentially one unknown systems of equations with literal or algebraic coefficients. They may recognize the power of algebra in this, but as the systems get larger, they may see that algebraic formulas derived become unwieldy or awkward. So they are cases in which numerical solutions are more convenient.

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4. In the subfolder Gaussian Elimination three forms elimination methods for solving systems (sets) of linear equations in to unknowns are introduced.
   • Substitution
   • Comparison
   • Equation (or Row) Addition and Subtraction, as is or after multiplication.

   Students have to earn all three. Student are told to watch for situations in which one requires less work than the others. As with the triangular systems, answers need to be checked by ensuring that all the equations in the origin systems are satisfied.

   Gifted students invited to solve systems of equations of equations in two unknowns with literal or algebraic coefficients instead of numerical ones. They may recognize the power of algebra in this, but as the systems get larger, they may see that algebraic formulas derived become unwieldy or awkward. So they are cases in which numerical solutions are more convenient.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science