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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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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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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.
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Chapter 15
Solving Linear Equations
Previous Chapter: Algebra
versus Arithmetic Solutions in forward and backward use of the Compound
Interest Formula
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.
1 One Unknown
1.1 First Example
When we let x = 5, we have 2x = 10 and 4x ¹
15. Suppose now we forgot the value of x which made 2x = 10, could
we find the value of x from the equation 2x = 10? The answer is
yes. We can solve for the unknown or forgotten value of x as follows:
In this solution, we used the property [(ab)/(b)] = b with
the role of a played by x and the role of b played by 2.
This gives the first equality. The second equality follows from assumption that
2x = 10. The latter allows 2x to be replaced by its value 10.
Another way to look at this solution is to say
Therefore
Hence
The manipulation process here creates new equalities from previous ones until an
expression
appears. How we get find the value of x from an equation involving x
or other unknowns is a matter of taste.1
1.2 Second Example
Problem: Find the value of x which satisfies the
equation 7x+9 = 65.
Solution: The aim is to manipulate (or change or massage)
the given equation
to get a new one of the form
The first step is to subtract 9 from both sides. This gives
Some of you may know that 65-9 = 56. We could write
56 instead of 65-9. A next step to further isolate x
is to divide by 7 (or multiply by [1/7]) since [(7x)/7] = x. This
manipulation gives
Therefore
The isolation of x is complete. The solution is x = 8. To check
this, just in case we made a mistake, observe when x = 8, we have 7x+9
= 7·8+9 = 56+9 = 65. So the original the equation 7x+9 = 65 holds (is
satisfied, is true) when x = 8.
1.3 Third Example
Problem: Find the value of x which satisfies 5x+6
= 117.
Solution: The aim is to manipulate the given equation
to get a new one of the form
A first step is to subtract 6 from both sides. This gives
A next step to further isolate x is to divide by 5 (or multiply by [1/5])
since [(5x)/5] = x. This manipulation gives
Therefore
| x = |
111
5 |
= 22+ |
1
5 |
= 22 |
1
5 |
|
|
The isolation of x is done. The solution is x = [111/5]. To check
this, just in case we made a mistake, observe when x = [111/5], we get 5x+6
= 111+6 = 117.
The solution
| x = |
111
5 |
= |
111 ×2
5 ×2 |
= |
222
10 |
= 22.2 = 22 |
1
5 |
|
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can be written in several ways. Which way we prefer is a matter of taste.
Further sections of this chapter are given by the web pages
[15 Algebra Solutions] [15
Triangular Systems] [15 Making Triangular] [15
Equations With 3 Unknowns] [15 Rules and Advice]
Use the next and back buttons in these links to move between these sections
or to return this page.
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www.whyslopes.com
Volume 2, Three Skills for Algebra -
Preview, starter & further lessons for logic and algebra
to (i) improve work & study skills; (ii) to to ease or avoid
algebra (math) fears & difficulties; and (iii) to fill gaps in the
exposition of mathematics.
Foreword, Chapters and Appendices follow.
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
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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