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YOU are better than YOU think. Show
yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
Read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. |
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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.
| |
Intersection of Lines
When a point [x,y] is on two lines, it must satisfy the equations of
both. This observation leads to two "linear" equations in two
unknowns
ax + by = e
cx + dy = f
These simultaneous equations must have a unique solution if the lines are
different and not parallel. The equations of lines y = e can be rewritten
in the form ax+by = e with b = 1.
To find the intersection point of a line y = m1x + b1
and y = m2x + b2 , solve the equation m1x
+ b1 = m2x + b2 for x and then compute
y. The two simultaneous equations
y = m1x + b1 and
y = m2x + b2
can also be solved in any way you wish. Subtraction
of one from the other would eliminate y immediately.
Working with the set model of a line
In the algebraic view, we may say or define a LINE is the set of
solutions (x,y) to an equation y = mx +b. Then the two-point formula for
the coefficient m, here
m = (y2 - y1)/(x2 - x1)
= rise/run
is a consequence of the equation without any reference to lines drawn in the
plane.
By describing a LINE as a set of points satisfying an equation, we obtain a
numerical model for a geometric object, the physical line in the plane, with
which we work algebraically and exactly apart from the approximation we do with
fat points (dots) and thick lines in the plane. Drawing errors are thus removed.
Which is more reliable: Geometry with or without coordinates?
The solution and properties of equations in analytic geometry
provides a model in which reasoning with numbers and algebra exactly
replaces the suggestive geometric reasoning in which a single diagram represents
and approximates many.
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www.whyslopes.com
Analytic Geometry
& Functions, etc
Area Entrance
Entrance + Pages Below this page
Pages at Current Level (L) Numericall Intro
(L) Deriving Eq'ns
(L) Perpendicular Lines
(L) 3 Eqn Forms
(L) Algebraic View
(L) Finding Intersection Points
(L) Coordinate Only Geometry.
(L) Exercises
(L) Lines Summary
(L) Lessons Elsewhere
Area Entrance
(C) Complex Numbers
(FN) What are Functions?
(FN) Functions - More
SZM: Sign Analysis
(L) Lines Summary
(P) Polynomials (*,+,-)
(Q) Quadratics
(D) Simplify Square Roots
(T) Unit Circle Trig
Conic Sections
Links
More Links
Pages at Above this Page
Extras: Not all perfect.
Equal Sign Use/Abuse
Real Numbers
Say More Positive
Linear Inequalities
Triangle Inequality
Absolute Value |x|
|x| Eq'ns & Inequalities
Rectangular Coords
Shortest Path
Distance Formulas
Add & Multiply Points
Polar Coordinates
Radians
(A) Vectors
(A) Coordinate Arithmetic
(A) Navigation on Maps
(A) Addition Geometrically
(A) Rotation
(PT) Translations
(PT) Dilatations
PT: Rotations
Links to Site Pages outside this site area follow - co- and pre-
requisites.
Road
Safety Message
Easy Consequences of this (newest) Complex
Number. Starter Lesson follow below to provide an alternate
development of HS or college maths.
Vec & Cmplx No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint
Arithmetic Videos - Real Player Format
Decimal Addition Methods
Decimal
Subtraction Methods
Decimal
Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisor
Divisors
Least Common Multiples
Square Root
Simplification
Using formulas forwards
& Backwards - A unifying theme for algebra skill development - the 4th
skill in Volume 2, Three
Skills for Algebra!
What
is a Variable?
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