YOU are better than YOU think. Show yourself  how:  

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 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

After logic,   (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.

Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Chapter 15
Solving Linear Equations

Previous section: Linear Equations With 3 Unknowns

Here are some more examples in which we solve equations. Our aim is to become familiar or at ease with handling and manipulating equations. So we look at the algebraic solution of equations containing one or more unknown numbers.

The equal sign = means the expressions to the left and right of it have the same value.

6  Useful Arithmetic Rules & Advice

6.1 Useful Arithmetic Rules

²Talking about the preservation of equality can be a wordy exercise - to be done at least once, but not more than thrice.

When a, b and c are shorthand symbols or expressions representing real numbers, the following properties are useful in solving equations.

• If a = b, that is if a and b are two expressions which when evaluated should give the same number (or quantity) then we should have
  

  ac = bc
  a+c = b+c, and
  a-c = b-c.

  whenever we multiply, add or subtract by another number or expression c. In brief, equality should be preserved (kept) if what we do to one side, we also do to the other.²
• If a+b = c+b then a = c. This follows by subtracting b from both sides.
• If ab = cb and b is nonzero then a = c.
• If ab = cb and b is nonzero then a = c. This follows by dividing both sides by b or by multiplying both sides by [1/(b)] = 1¸b.
• If a = b and c is nonzero then 

  a c

  =

  b c

6.2  Advice

Seeing and solving equations with one or several unknowns should help build your confidence in your algebraic abilities. That is good provided you understand at least once why each step is justified. Once you have obtained this understanding, the arithmetic properties given above can be used to solve equations and manipulate expressions in a mechanical way. But before you reach this mechanical, thoughtless state for solving equations, you should try to understand at least once the justification we might have for the arithmetic properties identified above.

Suggestion: Ask a teacher for more examples. A few more examples can also be found in the VNR Concise Encyclopedia. In solving equations you should find that there is more than one path to get a solution. Each path is good. Just remember to check that what you think is a solution is indeed a solution and that the path you followed contains no errors - In long calculations errors may accumulate or cancel. The presence of errors suggests a flaw in your reasoning process which needs to be identified and corrected.

More Chapter Sections: [ Up ] [ 15 Algebra Solutions ] [ 15 Triangular Systems ] [ 15 Making Triangular ] [ 15 With 3 Unknowns ] [ 15 Rules and Advice ]

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2. Three Skills for Algebra 

Foreword, Chapters 
& Appendices 

Foreword
1. Introduction
2. Implication Rules
3. Chains of Reason
4. Romeo and Juliet
4. Induction Mathematical
5 Knowledge Islands
6  Old Language
7  Arith Skill Check
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable
9. Algebra Talk
10 Two More Skills
11 Why Shorthand
12 Shorthand Usage
13 What's Next
14 Compound Interest
15 Linear Equations
PS I.  Distributive Law
PS II. Polynomials
16 Painless Proofs
17 Pythagoras
18 Rules of Algebra
19  Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums
23 Summation Notation
24 Your Money
25 Induction & Recursion
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
A. Advice For Learning

Real Player Videos Perfect arithmetic skills with whole numbers & fractions after or besides chapters 1 to 14. Arithmetic Videos Summary Addition with Decimals Subtraction with Decimals Multiplication with Decimals Fraction Arithmetic Recognizing Primes Long Division for Decimals Square Root Simplification Greatest Common Divisors Least Common Multiples

Words Before Symbols: 
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent or Independent
Variable, a Matter of Choice

Complex number: starter lesson  

Solving Linear Equations:

A. Letters and Lengths
B. & C. Solving Linear Eq'ns
with stick diagrams.

(i) x + 20 = 29
(ii) 2x + 5 = 20
(iii) 3x + 10 = 32
(iv) 5a + 16 = 3a+ 24
(v)  (½)x + 8 = 24½
(vI)  (¾)a + 16 = (¼)a+ 24
(vii) (¾)q + 17 = 32
(viii) 13 =[2/3]x +7 twice
(x) Animated Examples
(i) Integral Coefficients (A)
(ii) Integral Coefficients (B)
(iii) Fractional Coefficients
(iv) With Parameters

Problem Solving with Linear
Equations in one or many
unknowns, and in essentially 
one unknown - Symbols before
words. 

C. Solving Linear Eq'ns 
without
Stick Diagrams
D. Problems in 
essentially one unknown
E: 2D Systems - Sub Methods.
F. Larger Systems

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