Algebra 15 Algebra Solutions from Arithmetic Pattern

2 Algebraic Shorthand Solution

In a play or movie, the roles are more important than the actors, stars excepted. That is, any role can be played by any actor. But after the cast is selected, each role is usually played by only one actor, and each actor usually plays only one role. Once a play (or scene) is finished, the actors can take roles in another play (or scene). Likewise, in algebra, we have choice in the selection of the shorthand notation in which a problem or its solution is posed. But after the selection, the choice should be fixed at least temporarily. Once the problem and solution have been treated, the shorthand in it can be recycled in another plot.

2.1 Third Example Revisited

The role of x in the third example can be played by any other letter, for instance y. We will repeat the third example problem with y in place of x. (This is mathematics ad nauseum.)

Problem: Find the value of y which satisfies 5y+6 = 117. (This problem is identical to the previous one, except the shorthand symbol for the forgotten or unknown number is now the letter y instead of the letter x. The solution is identical. It is given or repeated next. Excuse the repetition, but you must see that it is a repetition.)

Solution: The aim is to manipulate the given equation

5y+6 = 117

to get a new one of the form

y = a numerical value

A first step is to subtract 6 from both sides. This gives

5y = 117-6

A next step to further isolate y is to divide by 5 (or multiply by [1/5]) since [(5y)/5] = y. This manipulation gives

y = 5y
5 = (117-6)
5

Therefore

y = 111
5 = 22+ 1
5 = 22 1
5

The isolation of y is done. The solution is y = [111/5]. To check this, just in case we made a mistake, observe when y = [111/5], we get 5y+6 = 111+6 = 117.

2.2 An Algebraic Pattern

Each of the above examples has the form ax+b = c in which the numbers a, b and c are given, and x is initially unknown. In the first example, the roles of a, b and c were played or given by 7, 9 and 65. That gave the equation 7x+9 = 65. In the second example 5x+6 = 117, the number 5 is used in place of a, the number 6 plays the role of b and the number 117 is given by c.

General Problem: Find x if ax+b = c.

ALGEBRAIC SHORTHAND SOLUTION. We follow the pattern set in the previous examples. First we subtract b from both sides of the equation ax+b = c. This gives

ax = c-b

Next, we observe if a is nonzero,

x =

(c-b)

Thus the formula for x is

x =

(c-b)

This gives a recipe for x no matter what values of a, b and c are given in the problem: find x if ax+b = c. The formula can be used when a š 0. Division by zero is not permitted or done in arithmetic. It is not possible.

Check: When x = [(c-b)/(a)], we see ax+b = aˇ[(c-b)/(a)] = (c-b)+b = c as hoped.

The recipe

x =

(c-b)

describes and gives the solution to many problems of the form ax+b = c.

Problem	Solution
ax+b = c	x = [(c-b)/(a)]
5x+6 = 65	x = [(65-6)/5]
7x+9 = 117	x = [(117-9)/7]
7y+9 = 117	y = [(117-9)/7]
123x+456 = 12067	x = [(12067-456)/123]
100x+(-20) = 800	x = [(800-(-20))/100] = [(800+20)/100] = 8.2
100x-20 = 800	x = [(800-(-20))/100] = [(800+20)/100] = 8.2
[4/5]x+4 = 10.2	x = [(10.2-4)/([4/5])]
3z+7 = 19	z = [(19-7)/3]

The formula x = [(c-b)/(a)] describes and gives a solution to many problems of the form ax+b = c. We can further use this recipe without repeating each time, the reasoning that led to it.

EXTRA. The above formula for x can be used to solve the equation ax-d = c by putting b = -d. The equation ax-d = c can be rewritten as ax+(-d) = c since subtraction of d can be replaced by the addition of the number -d.

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

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Chapter 15 Solving Linear Equations

2 Algebraic Shorthand Solution

2.1 Third Example Revisited

2.2 An Algebraic Pattern

Chapter 15
Solving Linear Equations