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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.
| |
Three Forms of an Equations for Straight Lines
1. Point Slope Form
Previously, we saw an
equation for a line with slope (rise to run ratio) m is given by
y-y1 = m·(x-x1)
if the line passes through or includes a point (x1,y1).
An equation in this form provides the a Point Slope equation for the line.
For each point on the line (x1, y1),
there is a point-slope equation. But given (x1, y1), we
can take the equation y - y1 = m(x-x1) and express
y in terms of x. That operation yields y = mx +b where b = y1-
mx1. Now x = 0 in y = mx +b implies y = b. So the
point (0,b) gives the lines intersetion with the y-axis. Since the line is not
vertical, it has only one point of intersection. So b b = y1-
mx1 is unique and independent of the choice of (x1,
y1). The convention that we rewrite the point-slope form
y - y1 = m(x-x1)
as y = mx+b by solving for y provides the unique
y-intercept slope equation of the line L.
2. Slope-Intercept Form
The slope intercept form of equation for a line y = b +m·x
or y = m·x + b. Clearly, x = 0 implies y = b.
Note: Some texts call this the functional form since the above
implies y = f(x) where f(x) = m·x + b.
The point-slope equation of a line, that is, y-y1
= m·(x-x1) implies y = y1+m·(x-x1).
The latter in turn implies y = mx + y1-m·x1 = mx +
b on putting b = y1-m·x1. That is an algebraic
detail. Suffice it to say that when numbers are present, the
point-slope form (and two-point form) of the equation for a line imply the form
y = mx + b for some number b that can be calculated (exactly svp) from the
numerical coefficients in y = y1+m·(x-x1).
In the case [x1, y1] = [0, b] is
the y-intercept, equation y = y1+m·(x-x1)
becomes the slope intercept form of equation for a line y = b +m·x
or y = m·x + b. Explain why.
3. Two Point Form
The value of the slope m can be obtained from the two point formula
| m = |
Dy
Dx |
= |
y2-y1
x2-x1 |
= |
rise
run |
if a second point (x2,y2) is available.
The substitution of the latter expression into y-y1
= m·(x-x1)
gives the two point form equation for a line,
| y-y1 |
= |
[ |
y2-y1
x2-x1 |
] |
·(x-x1) |
Cosmetic Convention - Keeping Up Appearances
In answering questions, rewrite any equation you obtain for a non-vertical
line into a slope intercept equation. After an equation of a line is
written or given in form y = m·x + b, the coefficient of x
gives m and the constant term b is the y-intercept, that is the value of y when
x = 0.
Exercise 1. For the equation y = 3 x +8,
show for each unit change in x, there is a change of m = 3 in y. If the run is
1 then then the rise is m*1 = 3*1 = 3 in this case.
Exercise 2. For the equation y = m x +b,
show for each unit change in x, there is a change of m = 3 in y.
Exercise 3. If m = r/s in the equation y =mx+b, show
for each change of s in x, there is a change of r in y. Hint: All
follows from r = m*s where m = the rise over run proportionality
constant.
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www.whyslopes.com
Analytic Geometry
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
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(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
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Extras: Not all perfect.
Equal Sign Use/Abuse
Real Numbers
Say More Positive
Linear Inequalities
Triangle Inequality
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Shortest Path
Distance Formulas
Add & Multiply Points
Polar Coordinates
Radians
(A) Vectors
(A) Coordinate Arithmetic
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(A) Addition Geometrically
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PT: Rotations
Links to Site Pages outside this site area follow - co- and pre-
requisites.
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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
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Divisors
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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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