Algebra 15 Algebra Solutions from Arithmetic Pattern

www.whyslopes.com Appetizers and Lessons for Mathematics and Reason a calculus and preparation for calculus website, etc.
Online Volumes (Book Orders) 1, Elements of Reason. 1A. Pattern Based Reason 1B. Math Curriculum Notes 2. Three Skills for Algebra 3. Why Slopes & More Math Mathematics Course Designers: LAMP offers food for thought.	More Site Areas 1. Help Your Child or Teen Learn 2. Solving Linear Equations 3. Fractions Ratios Rates Proportions & Units 4. Euclidean Geometry 5. Analytic Geometry/Functions 6. Number Theory. 7. More Calculus	More Site Areas 8. Complex Numbers 9. Qc Maths Education 10. Secondary IV(?) maths 11. Real Analysis 12. LaTeX2HotEqn: 13. Electric Circuits Etc 14. Français 15. Algebra, Odds & Ends, Etc	More Site Areas 16. Math Education Essays 17. Telling & Working with Time 18. Maps, Plans & Drawings 19. Quantitative Skills for home, shopping and work 20. Statistics Useful, or Not. Try the Twiddla Whiteboard to work online with others.
\|\|Définition d'une variable \|\| Algèbre \|\| Arithmetique \|\| Logique \|\|La raison basée sur les règles et modelés\|\| [Site Entrance & Hub][ Up ] [ Up ] [ Next ][Site Exit]

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 ¹ 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 = a·[(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.

Chapter 15 Solving Linear Equations

2 Algebraic Shorthand Solution

2.1 Third Example Revisited

2.2 An Algebraic Pattern

Chapter 15
Solving Linear Equations