How to Solve Word Problems Programmatically

Chapter

13

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Chapter 13.  Simplifying Expressions

 

 

 

 

13.1

Substituting Numbers

13.2

Evaluating Numbers

13.3

Order of Operations

13.4

Basic Algebra Rules

13.5

Using Rules to Simplify Expressions

13.6

Deriving Other Rules

13.7

Rules for Factoring

13.8

Rules for Exponents

13.9

Activities and Explorations

 

 

 

13.1

Substituting Numbers

 

 

Expression

Substitution Numbers

Results

(a+b)*c

a=4, b=2, c=3

 

a*q2^q1

a=4, q1=2, q2=3

 

y/-x^2

y=8, x=2

 

third/first*second

first=2, second=3, third=12

 

a+b*c

a=4, b=2, c=3

 

q-r+s

q=6, r=2, s=3

 

a+b+c

a=2, b=4, c=5

 

5+4*x-3*x^2

x=2

 

a+x/b-x/c

a=4, b=2, c=3, x=6

 

3*(x+2)-x^2

x=-3

 

 

Substitute the numbers into their corresponding expressions and calculate what you think is the value of the expression. Do not worry about right or wrong. The purpose of this exercise is to help you become aware of the conventions that mathematics uses to evaluate expressions.

 

Be aware that we use the sign "-" in two different ways, as a "negative" sign and as a "subtraction" operator. The subtle difference between these two uses can cause problems if you do not take care. One of the basic rules of algebra (discussed later) defines subtraction as adding the additive inverse (indicated by the negative sign) of a number: a-b=a+(-b).

 

 

13.2

Evaluating Numbers

 

 

Expression

Substitution Numbers

Results

(a+b)*c

a=4, b=2, c=3

(4+2)*3=6*3=18

a*q2^q1

a=4, q1=2, q2=3

4*(3^2)=36 or (4*3)^2=144

y/-x^2

y=8, x=2

8/-(2)^2=-2 or 8/(-2)^2=4

third/first*second

first=2, second=3, third=12

(12/2)*3=18 or 12/(2*3)=2

a+b*c

a=4, b=2, c=3

2+(4*3)=14 or (2+4)*3=18

q-r+s

q=6, r=2, s=3

(6-2)+3=7 or 6-(2+3)= 1

a+b+c

a=2, b=4, c=5

(2+4)+5 = 2+(4+5)=11

5+4*x-3*x^2

x=2

multiple possibilities

a+x/b-x/c

a=4, b=2, c=3, x=6

multiple possibilities

3*(x+2)-x^2

x=-3

multiple possibilities

Some of these examples have more that one result that seem reasonable. Shall we vote on them democratically? Mathematics resolves this by a set of conventions (the language syntax) one of which specifies the "order of operations". Try putting these examples into your calculator and make sure you know the proper keystrokes to get right answer as determined by the following order of operations.

 

 

13.3

Order of Operations

 

 

Sign

Operations

Examples:

 

( )

paranthesis

(a+b)*c

(4+2)*3=6*3=18

^

exponent

a*q2^q1

4*3^2=36, not (4*3)^2=144

-

negative

y/-x^2

8/-(2)^2=-4, not 8/(-2)^2=4

/

division

third/first*second

12/2*3=18, not 12/(2*3)=2

*

multiplication

a+b*c

2+4*3=14, not (2+4)*3=18

-

subtraction

q-r+s

6-2+3=7, not 6-(2+3)= 1

+

addition

a+b+c

(2+4)+5 = 2+(4+5)

 

The order in which these operations appear in this table, determines which of these operators we do first. Parenthesis always goes first; so when you want an expression to calculate first, put it in parenthesis.

 

Traditional notation used in textbooks sometimes hides operations (especially multiplication, exponentiation and parenthesis). By converting traditional notation to a text-string notation used by many computer software applications, we see explicit symbols for each operation and can determine more readily the sequence of operations calculated.

 

These rules establish the conventions upon which we read a mathematical statement. As demonstrated by the examples, understanding which operation we do first can make or break an effort to solve a word problem. Now even the most complicated expressions end up with only one accepted value:

 

Expression

Substitution Numbers

Results

5+4*x-3*x^2

x=2

5+4*x-3*x^2=5+8-12=1

a+x/b-x/c

a=4, b=2, c=3, x=6

4+6/2-6/3=4+3-1=6

3*(x+2)-x^2

x=-3

3*(-3+2)-(-3)^2=-3-9=-12

 

13.4

Basic Algebra Rules

 

 

Rule Name

Statement

Example

Simplify:

(Addition)

 

 

 

Identity

a+0=a

2/3+3/2-6/4

 

 

 

Inverse

a+(-a)=0

5+(1-6)

 

Commutative

a+b=b+a

-4+5/2+4

 

 

Associative

(a+b)+c=a+(b+c)

(5+-6)+6

 

 

Definition of Subtraction

a-b=a+(-b)

4-(-6)

 

 

 

 

 

(Multiplication)

 

 

 

Identity

a*1 = a

(4-8/4)*(3-2)

 

 

Inverse

a*(1/a)=1

6*8*(1/6)

 

 

Commutative

a*b=b*a

5*9/5

 

 

Associative

(a*b)*c=a*(b*c)

(3*4/5)*5

 

 

Definition of Division

a/b=a*(1/b)

4/(1/4)

 

 

 

 

 

(Connects Them)

 

 

 

Distributive

c*(a+b)=c*a+c*b

12(3/4+5/12)

 

 

 

These 11 rules form the foundation for algebra. Just like geometry can be constructed from just 5 rules (Euclid's postulates), most of algebra can be derived from the above rules. Go ahead and simplify each of the above examples to illustrate the corresponding rule (and sometimes use some of the other rules.)

 

 

 

 

 

 

13.5

Using Rules to Simplify Expressions

 

 

Rule

Statement

Example

Results:

(Addition)

 

 

 

Identity

a+0=a

3/4+3/2-6/4

 

 

3/4

Inverse

a+(-a)=0

5+(1-6)

0

Commutative

a+b=b+a

-4+5/2+4

 

5/2

Associative

(a+b)+c=a+(b+c)

(5+-6)+6

 

5

Definition of Subtraction

a-b=a+(-b)

4-(-6)

10

 

 

 

 

(Multiplication)

 

 

 

Identity

a*1 = a

(4-8/4)*(3-2)

 

2

Inverse

a*(1/a)=1

8*6*(1/6)

 

8

Commutative

a*b=b*a

5*9*(1/5)

 

9

Associative

(a*b)*c=a*(b*c)

(3*4/5)*5

 

12

Definition of Division

a/b=a*(1/b)

5*9/5

9

 

 

 

 

(Connects Them)

 

 

 

Distributive

c*(a+b)=c*a+c*b

12(3/4+5/12)

 

14

 

Did you get these results when you evaluated these expressions? These rules allow us to simply expressions with variables just as easily as we evaluated these expressions. To illustrate this, go ahead and use these basic rules to simplify the following expressions.

 

Example

Results:

 

 

3/4+6*x/2-3*x

 

 

 

5+(x-6*x)*(1/x)

 

-4*x+5*x^2/2+4*x

 

 

(5*x^2+-6*x)+6*x

 

 

4*x-(-6*x)

 

 

 

(4*x-2)*(3*x+1-3*x)

 

 

8*x*(6+x)*(1/(6+x))

 

 

5*x^3*9*(1/(5*x^3))

 

 

(3*x*4/5)*5

 

 

5*x^3*9/(5*x^3)

 

 

 

12(3*x/4+5*x/12)

 

 

 

 

 

Check out the results of simplifying these expressions that containing variables using (at least) the rule that it appears with.

 

Rule

Example

Results:

(Addition)

 

 

Identity

3/4+6*x/2-3*x

 

 

3/4

Inverse

5+(x-6*x)*(1/x)

0

Commutative

-4*x+5*x^2/2+4*x

 

5*x^2/2

Associative

(5*x^2+-6*x)+6*x

 

5*x^2

Definition of Subtraction

4*x-(-6*x)

10*x

(Multiplication)

 

 

Identity

(4*x-2)*(3*x+1-3*x)

 

4*x-2

Inverse

8*x*(6+x)*(1/(6+x))

 

8*x

Commutative

5*x^3*9*(1/(5*x^3))

 

9

Associative

(3*x*4/5)*5

 

12*x

Definition of Division

5*x^3*9/(5*x^3)

9

 

 

 

Distributive

12(3*x/4+5*x/12)

 

14*x

 

 

 

 

 

 

 

13.6

Deriving Other Rules

 

 

(a+b)^2=a^2+2*a*b+b^2

 

or equivalently

 

(a+b)2=a2+2*a*b+b2

Sum-Squared Rule

 

One nice feature of mathematics comes from the fact that if you cannot remember a formula (like this expansion formula), you can often remember related formulas and derive the forgotten one. The 11 basic rules presented above form the foundation upon which we derive most of the other algebra rules. Let us use them to derive (prove) the above sum-squared rule.

 

Rule

Reason

(a+b)^2=

 

(a+b)*(a+b)=

Definition of exponent

(a+b)*a+(a+b)*b=

Distributive rule (from left)

a*a+b*a+(a+b)*b=

Distributive rule on left multiplication

a*a+b*a+a*b+b*b=

Distributive rule on right multiplication

a*a+a*b+a*b+b*b=

Commutative rule on second term

a*a+2*a*b +b*b=

Associative rule on two middle terms

a^2+2*a*b +b^2

Definition of exponent

 

13.7

Rules for Factoring

 

 

Rule

Example

Result:

(a+b)2=a2+2*a*b+b2

(5+x)^2

 

(a-b)2=a2-2*a*b+b2

(1-x)^2

 

a2-b2=(a+b)*(a-b)

1-x^2

 

(a*x+b)(c*x+d)=a*c*x2+(a*d+b*c)*x+b*d

(2*x-3)*(5*x+4)

 

 

 

 

 

 

 

 

The last rule generalizes the factoring of a quadratic expression. We often use it in reverse where we have the expression on the right of the equal sign and then determine the left side of the equation. These techniques (usually covered in a second year algebra course) take some practice, but are helpful in simplifying fractions of expressions like the last added example. The website http://icm.mcs.kent.edu/research/facdemo.html allows you to enter a polynomial and get factors, if they exist. Try entering 10*x^2-7*x-12 and see what you get.

 

Did you get the following results for the above table of examples?

 

Example

Results:

 

 

(5+x)^2

25+5*x+x^2

(1-x)^2

1-2*x+x^2

1-x^2

(1+x)(1-x)

(2*x-3)*(5*x+4)

10*x^2-7*x-12

(10*x^2-7*x-12)/ (2*x-3)

(5*x+4)

 

 

 

http://www-math.mit.edu/18.013A/tools/tools04.html (Visual blocks to represent factors)

 

 

 

13.8

Rules for Fractions

 

 

Rule

Example

Result:

(a/b)*b=a

((x+7)/4)*4

 

a/b+c/d=(a*d+c*b)/b*d

 

x/2+y/6

 

a/b-c/d=(a*d-c*b)/b*d

1/3-2/A

 

(a/b)*(c/d)=(a*c)/(b*d)

(((x-5)^2)/5)*(5/(x-5))

 

(a/b)/(c/d)=(a/b)*(d/c)

(4/c)/(c/2)

 

(a/b)^n = a^n / b^n

(x/3)^n

 

 

Discussion of fraction rules can be found at

http://library.thinkquest.org/20991/textonly/alg/frac.html

 

 

 

13.9

Rules for Exponents

 

 

Rule

Example

Simplify:

ax*ay=ax+y

(2+y)5(2+y)-4

 

(ax)y=ax*y

((2 (1/x+1))2*x

 

(a*b)x=ax*bx

(1+x)3 * (2/(1+x))3

 

ax/ay=ax-y

z5/z3

 

a-x=1/ax

(2+y)4*(1/(2+y) 3)

 

(a/b)x = ax/bx

(1-x2)3/(1-x)3

 

 

 

 

 

13.10

Activities and Explorations

 

 

 Activities:

 

Interactive Review of Some Algebra Rules

http://www.sosmath.com/algebra/fraction/frac8/frac8.html

 

College Algebra Online Tutorial

http://www.ohaganbooks.com/ThirdEdSite/tutindex.html

 

Find Factors of Some Polynomials

http://icm.mcs.kent.edu/research/facdemo.html

 

 

Explorations:

 

Algebra As a Language

http://www.psychstat.smsu.edu/introbook/sbk05.htm

 

Review Fraction Rules

http://library.thinkquest.org/20991/textonly/alg/frac.html

 

  

Exercises:

 

Do Related Problems In Your Textbook

 

 

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Copyright 2004  Dr. Ranel E. Erickson