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More Problems On Solving Linear Equations, Etc.
Show work. Use Proper Notation. Check your answers if time permits.
1. (10 points) Compute: Express as a single, simplified,
improper fraction:
A= [
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5
4
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¸
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35
24
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] ¸
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2
7
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B=
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4
5
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[
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5
4
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+
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35
24
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+
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2
7
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]
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2. (10 points) Solve
x - 2y = 4
-3x + y = 3
3. (10 points) Solve
2x + 3y + 4z = 21
8y + 2z = 26
5y = 15
4.. (10 points) Solve - note minus sign in each
equation.
-x + y + z = 10
x - y + z =
8
x + y - z = 6
x + y + z - w =14
5.. (4 points) (i) Show by substitution that the
formulas
x =
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dg
ad-bc
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and y =
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-cg
ad-bc
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satisfy (solve) the system
ax + by = g
cx + dy = 0
(2 points) (ii) Give values for coefficients a, b, c,
d, and g to make the system in (i) the same as
4x + y = 10
x + 4y = 0
(4 points) (iii) Use the formulas in (i) to solve
the system in (ii)
6. (Bonus 4 points) Explain (or show) how the law of
signs for real numbers follows from the rule: add the angles, multiply
lengths for multiplication of complex numbers or points in the
plane.
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Professor Whyslopes:
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Site value lies in the difference between its
ideas and yours.
-
If one site explanation is not to your
liking, try another. Each one is different.
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Two gaps
-
The Old Algebra Gap: Algebra appears with
too few words of explanation in high school and college
mathematics. Online Volumes 2 and 3 offer remedies. Chapters
8 to 12 in Volume 2 put more words into the explanation and
comprehension of algebra. Chapter
14 in Volume 2 with its explicit discussion of the direct and
indirect use a formulas identifies a unifying theme for
mathematics and logic - all rules and patterns will be used
forward and backwards. Chapters 2
to 6 and 12 to 18 in Volume 3 may further ease or avoid the
very challenging use of algebra in the high level mathematics:
calculus. Calculus requires earlier high school
mathematics at full strength: (i) This logically complete but
long lesson on complex numbers shows how
to simplify the senior high school exposition of circular trig
functions upto to formulas in the plane for vectors dot and
cross-products. The lesson provides the route that would have
been taken in course design if the key element of the lesson, a
December 2009 invention, had been available in the 1950s. For
further algebra skill development. See the site coverage of
fraction with units,
proportionality, ratios and rates,
polynomials,
quadratics
functions and
straight line slopes and equations.
-
The Arithmetic Gap: An exact and efficient
mastery of arithmetic with decimals and fractions is best
(required) for the high level study of mathematics alone and in
science, technology and business. Pages here on arithmetic with decimals and
integers, on
fractions and solving linear
equations with
fractional operations on stick diagrams may help fill the
gap. That exact and efficient command should be obtained in the
last years of primary school and the first years of secondary
school.
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Skill mastery in mathematics has to be
seen to believed. To that end, learn or teach how-to write
and draw the steps in mathematical figuring or reasoning
clearly. Do not try to save space by doing a sequence of step
in one place. Instead, do or record the steps in sequence on
a separate lines to make each step obvious and
verifiable.
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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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