Step 3 Easy systems in 2 or more unknowns
1 Essentially One Unknown
2 Essentially one exercises - three with solution
3 Solving triangular system example
4 Solving a triangular system exercise
Notes
In this folder Easy systems in 2 or more unknowns, the easy
systems are provided by (i) groups of equations in essentially one
unknown and systems of equations that are triangular - or become so after
a changing the order of the equations. Students may be surprised in being
informed that a single letter has or should have one and only one value
in the "simultaneous" equations forming a systems. That is contrary to
their experience in solving linear equations - one isolated equation at a
time.
In triangular systems, one equation by itself gives the value of one
unknown. Then another equation by itself gives the value of a second
unknown. Whence the unknowns can be made known one at a time, one after
another. Recognizing triangular system and how to solve also the set
the stage for transforming general systems into essentially triangular
form. Answers need to be checked by ensuring that all the equations
in the origin systems are satisfied.
In systems in essentially one unknown, all unknowns are expressed in
terms of one - the key or essential unknown. By one or more
substitutions, a single equation in the key or essential unknown results.
Solution of the latter equation then gives the value of the key or
essential unknown. The derivation of the latter equation forces an
operational if not formal command of associative laws for multiplication
and/or distributive laws for multiplication over addition. Once the value
of the key or essential unknown is found, the values of the other
unknowns can be obtained. Answers need to be checked by ensuring that all
the equations in the origin systems are satisfied.
The substitution operations which turn one of the equations into a single
equation in one unknown also provide another partial model for solution
of linear systems in two or more unknowns. Altogether, the solution of
systems in essentially one unknown helps build arithmetic and algebraic
skills and confidence.
Many of the harder word problems in junior high school mathematics may be
cast as systems of equation in essentially one unknown, and then solved
algebraically. The discussion here of systems of in essentially one
unknown makes that easier. Alternatively, apart from the mastery of such
systems, students can endeavour to express all numbers and quantities in
a given word problem in terms of one unknown in a way that leads to a
single equation in the latter. On page 77 in the book
Problem Solving Through Recreational Mathematics, Averbach and
Chern, year 2000 edition,
solves an simple problem in three unknown ages with both approaches, in a
side-by-side two column format, given for the sake of comparison.
Opinion: The mechanical formulation of junior high school word
problem as a system of linear equations, the form of which identifies the
key or essential unknown, provide the simplest approach to such problems.
The treatment here of both kinds is accompanied by instruction on how to
check solutions. Here again if the check fails, the error or errors lies
between the first line of the solution and the last line of the check.
Gifted students can be invited to solve triangular and/or essentially one
unknown systems of equations with literal or algebraic coefficients. They
may recognize the power of algebra in this, but as the systems get
larger, they may see that algebraic formulas derived become unwieldy or
awkward. So they are cases in which numerical solutions are more
convenient.
The literal or algebraic solutions of equations is introduced further in
a small example in Chapter 10 and in many examples in Chapter 14 of site
Volume 2, Three Skills for Algebra. Chapter 14 in particular emphasizes
the forward and backward use of a formula - the compound interest
formula, while contrast arithmetic (a.k.a numerical) and algebraic (a.k.a
literal) solutions. The careful forward and backward use of rules and
formulas, proportionality relations included, represents a unifying
thread for mathematical and scientific subjects in secondary and college.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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