Equivalent Systems and Implied Systems
This lesson requires mastery of implication rules and allied concepts
from logic. The initial chapters
offers a light and enterntaining introduction to logic and how it
divides "deductive" knowledge. Reasons for reading them are given in
the next chapter of this online article: Secondary Mathematics, a
Practical Path, the article before you.
One equation in one unknown - Example
A second equation is implied by first if a solution of the first is also
a solution of the second. Chains of implied equations provide solutions
to an initial equation. For example, suppose we want to solve $3x+5 =
35$. Suppose x is a solution. Then we may write the following chains of
implications.
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If x is a solution of $3x+5 = 35$ then x is a solution of $3x=10$.
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If x is a solution of $3x = 30$ then $x=10$. The latter statement is
also an implied equation.
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If $ x = 10$ then $3x+5 = 35$ by inspection. So $3x+5=10$ is implied
by $x=10$
In the foregoing, the three equations [a.] $3x+5 = 35,$ [b.] $ 3x=30$ and
[c.] $x=10$ imply each other directly, or though longer chain of reasons.
The equations are equivalent. They have the same solution. Here $x=10$ is
read as giving the value of x and being an equation as well.
Quadratic Example
The equation $x^2=9$ has two solutions $x=3$ and $x={}^-3$. Here the
first equation $x^2=9$ is implied by the equality $x=3$ but is not
equivalent to the equality $x=3$ because $x={}^-3$ also implies $x^2=9$.
However, the equation $x^2=9$ implies $0 = 9 -x^2$, and vice-versa.
Moreover $x^2-9 = (x-3)(x+3)$ is a computational identity - why is
explained in the site lessons on quadratics. Therefore $0 = 9 -x^2$
implies and is implied by $0= (x-3)(x+3)$. By the zero product law, the
latter implies $x=3$ or $x=-3.$ Thus the single equation $x^2=9$ implies
and is implied by the pair of alternatives $x=3$ or $x=-3,$ alternatives
that mutually exclusive. So the single equation is equivalent to a pair
of alternative equations.
System of Linear Example
Suppose [x,y] is a solution of the system of equations \begin{eqnarray*}
15 &=& 5x + y \\ -9 &=& x - y \end{eqnarray*} Adding the
second equation to the first in accordance with the equality implication
pattern implies [x,y] must also be a solution of \begin{eqnarray*} 6
&=& 6x \\ -9 &=& x - y \end{eqnarray*} Mutiplying both
sides of the new first equation by a sixth $\frac16$ by the equality
implication pattern implies [x,y] must also be a solution of
\begin{eqnarray*} 1 &=& x \\ -9 &=& x - y \end{eqnarray*}
Multiply the second equation by -1 by the equality implication pattern
implies [x,y] must also be a solution of \begin{eqnarray*} 1 &=&
x \\ 9 &=& -x + y \end{eqnarray*} The distributive law was
employed to simplify the right hand side of the second equation. Now add
the first equation to the second to get \begin{eqnarray*} 1 &=& x
\\ 10 &=& y \end{eqnarray*} It easily seen that the last system
of equation gives two values of [x,y] which satisfy the original
equation. So the last system implies the first system and all following
systems via chains of reason. Each system implies the next. So all the
systems are equivalent. That implies there is only one solution.
Another way to the see the equivalent of each system with the next is to
observe that each system also implies the first. Let us see how. Assume
\begin{eqnarray*} 1 &=& x \\ 10 &=& y \end{eqnarray*}
Subtract the first equation from the second to imply the previous
\begin{eqnarray*} 1 &=& x \\ 9 &=& -x + y \end{eqnarray*}
Multiply the second equation by -1 to imply \begin{eqnarray*} 1
&=& x \\ 9 &=& -x + y \end{eqnarray*} Multiply the first
equation by 6 to obtain \begin{eqnarray*} 6 &=& 6x \\ 9
&=& -x + y \end{eqnarray*} Add the second to the first to obtain
the original equation. \begin{eqnarray*} 15 &=& 5x + y \\ -9
&=& x - y \end{eqnarray*} Because all the implications in the
chain of steps that led to the solution are reversible, each system in
the chain is equivalent to the previous one. Thus it should be no suprise
that the last system \begin{eqnarray*} 1 &=& x \\ 10 &=&
y \end{eqnarray*} solves the original system. That being said, mistakes
can be made in each step. No student is perfect. Hence all equations of
the original should be checked mentally when simple enought and on paper
otherwisze.
Systems in Essentially One Unknown
Suppose [p,q,r,s, t] are solutions of the system \begin{eqnarray*} 53
&=& 3p+4q+5(s+4)+4t \\ p &=& 2t+1 \\ q &=& 3t+1
\\ s &=& 4t+1 \end{eqnarray*} The substitutions \begin{eqnarray*}
p &=& 2t+1 \\ q &=& 3t+1 \\ s &=& 4t+1 \\
\end{eqnarray*} in the first equation \begin{eqnarray*} 53 &=&
3p+4q+5(s+4)+4t \\ \mbox{imply} &=& 2t+1 \\ 1 &=&
3(2t+1)+4(3t+1)+5((4t+1)+4)+4t \end{eqnarray*} The latter is an implied
equation for t. Because the substitutions are reversible, the system of
equations \begin{eqnarray*} 53 &=& 3(2t+1)+4(3t+1)+5((4t+1)+4)+4t
\\ p &=& 2t+1 \\ q &=& 3t+1 \\ s &=& 4t+1
\end{eqnarray*} is equivalent to the original. The last system is
equivalent to \begin{eqnarray*} 53 &=& 42t+ 32 \\ p &=&
2t+1 \\ q &=& 3t+1 \\ s &=& 4t+1 \end{eqnarray*} The
latter is a triangular system. The first equation gives $t=\frac12$ and
hence the full solution is \begin{eqnarray*} t &=&\frac12 \\ p
&=& 2 \\ q &=& 2+\frac12 \\ s &=& 3
\end{eqnarray*} While the latter equations provide the values of the
unknowns, and form a system that should be equivalent to the first, the
risk of mistakes requires the original system to be checked. The
verification or correction of results is left to the reader.
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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
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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:
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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
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gives boys and girls a head start. Good luck. At the other
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McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
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Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
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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
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Is your child able to add, subtract and multiply amounts
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work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
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Arithmetic
and Number Theory Skills
Algebra
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Geometry
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Algebra
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Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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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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