YOU are better than YOU think. Show yourself how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
|
Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work |
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
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;
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
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.
2D (Binary) Systems and Gaussian Elimination
Here are three elimination methods for solving systems (sets) of linear equations
- Substitution
- Comparison
- Equation (or Row) Addition and Subtraction, as is or after multiplication.
Learn all three, and watch for situations in which one requires less work than the others. The last method only works for linear systems (sets) of simultaneous equations.
Remember to check your answers by making sure all the equations in the systems being solved, the original one, are satisfied. If the check fails, you have made a mistake in obtaining the answer or doing the check. If you answer differ from someone else's, at least one of you has made a mistake, possibly both. Here there is a difference between the method for checking answers and the methods for obtaining answers.
Substitution (Replacement) Method
Example 1: The set or system of equations
x = 25 + 2y
x + 3y = 10
becomes a system in one unknown when we eliminate the x from the second equation by replacing x in the second by the expression 6 +2y. The second equation
x + 3 y = 10
then gives
(25+2y) + 3 y = 10
or
25 + 5y = 10.
So 5y = -15 and hence y = -3 (from -15 divided by 5)
Example 1 Revisited: Now consider the set or system of equations
x - 2y = 25
x + 3y = 10
In this system, neither x nor y is expressed in terms of the other. But the first equation implies x = 25 + 2y as before by adding 2y to both sides of it. So both equations together give the previous set of equations
x = 25 + 2y
x + 3y = 10
What we did here was express x in terms of the other unknown to allow us to convert the original system into a system with essentially one unknown. See the first example again.
Example 2: Animated
Example 3: Now lets us consider the third binary or two unknown system
-2x + y = 8
3x + 4y =21
Here the first equation implies y = 8 -2x. So the third system gives
y = 8+ 2x
3x + 4y =21
So we replace y in the second equation 3x + 4y =21 by the expression 8 + 2x that should have the same value as y. This implies
3x + 4 (8+2x) =21 or
3x + 32 + 8x =21 or
11x+32 = 21 or 11x = -11
So x = -1. Now y = 8 + 2x gives y = 8 +2(-1) = 6.
Note 4 (8+2x) = 32 + 8x due to the distributive law, namely a(b+c) = ab + ac whenever a, b and c are (real) numbers.
I will leave to you to check that
-2x + y = 8
3x + 4y =21
when x = -1 and y=6.
Example 4: Animated
www.whyslopes.com
Solving Linear Equations
|(Feb 14, 2005)A reference for solving linear equations and for recognizing word problems in essentially one variable. Skill in arithmetic with fractions is a must for algebra. .
Area Entrance Proper Use of Equal Sign A. Letters and Lengths B.. Solving Linear Eq'ns. C. Solving Linear Eq'ns D.Almost One E: 2D Systems - Sub Method. E: Continued E: Still More F. Larger Systems
Area Entrance (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
Up Proper Use of Equal Sign A. Letters and Lengths B.. Solving Linear Eq'ns. C. Solving Linear Eq'ns D.Almost One E: 2D Systems - Sub Method. E: Continued E: Still More F. Larger Systems
Arithmetic Videos
Decimal Addition Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisors
Least Common Multiples
Square Root Simplification
Site books and further webpages on learning and teaching mathematics and pattern based reason may develop critical thinking, improve reading and writing, and give a base for learning or teaching high school and college mathematics.
Great_Expectations: If you can learn to follow a multi-step methods in any subject precisely, you can do so in other subjects, as well.
Good news: Site pages identify what you need to study.
Bad news: Site pages do not explain everything
Worse news: Learning takes time, yours
Lesson Plans and Ideas for Teachers & Tutors:
Secondary I - fractions & allied concepts (decimals, percentages)