SUMMARY OF DISCUSSION WEEK 2 ALGEBRA(Variables)

CLICK THIS TO DOWNLOAD

Variables

A symbol used to represent a particular type of quantity

Ex.

s. = speed

d = distance

t = time

a = an integer

common variables = x and y

Expression

We can combined numbers and variables using fundamental operations of real numbers.

The resulting combination is what we call expression

Expression can be classified according to the number of variables

Expression with no variables are also known as constant

Expression is a combination of numbers and variables using the fundamental operations of real numbers.

5x²

3x/y

3xy

These expressions are what we call as terms

Terms is a combination of numbers and variables using multiplication, division and exponentation.

Terms can be considered as the building block of expressions.

Expression os a combination of terms using addition and substraction.

Terms has the ff. parts

2x

2 is numerical coefficient

X is literal coefficient

Literal coefficient refers to the variable part of a term while numerical coefficient refers to the numerical part of a term

Ex

3x²y

4  numerical is 4 and no literal

X   Numerical coefficient  is 1 and literal is x

FORMING EQUATION and INEQUALITY

We can combine two expression using the symbol of equality = to form a combination we call as equations.

Example

5x² + x = 1

12 –yz² = y

X –y = 2y

EQUATION is a combination of two expressions using the symbol of equality.

We can also combine two expressions using he symbols of inequality such as > < ≤ ≥  to form a combinations we call as inequalities.

Inequality is a combination of two expressions using the symbols of inequality such as > < etc

COMMUNICATING WITH VARIABLES

Substitution is a process of replacing a variable with a constant in a giver expression.

Truth value of an equation and inequality

A certain equation can be evaluated as true or false given a certain substitution. Consider the ff. equation

X² + x + 1 = 3

Performing substitution at x=2, we have

(2) ²+ 2 + 1 = 3

      7 = 3, equation is false

Performing substitution at x = 1

(1) ²+ (1) + 1 = 3

      3 = 3

SOLUTION

The solution of an equation is the set of numbers which make the equation true. The set of all solutions of an equation Is known as the solution set. For an equation 1 variable, the solution is just a constant

Consider the following examples:

	Equation	Solution Set
1	2x – y = 1	(1,1)
2	Y + X2 = 5	(2,1)
3	X3 = y	(2,8)

Where the first entry of the ordered pair stands for the value of the variable x and the second entry stands for the value of the variable y

These are verified by the following:

PROPERTIES OF REAL NUMBERS

An equation which is true for all real numbers can be considered as a property of real numbers. Consider the following properties:

Reflexive Property

Let x be a real number

Then,   x = x

Symmetric Property

Let x, y be real numbers

Then,

     If x = y then y =x

Transitive Property

Let x,y,z be real numbers

Then

     If x=y and y=z then x=z

These properties are also known as the property of equality. Consider the following properties

Commutative Property

Let x, y be real numbers

Then:

X + Y = Y + X

xy = yx

Associative Property

Let x, y, z be a real number

Then

X + (Y + Z) = (X + Y) + Z

X(YZ) = (XY)Z

Distributive Property

Let x,y,z be real number

Then

X(y + z) = xy + xz

(Y + z)x = yx = zx

Problem Solving

One of the fundamental skills to learn in problem solving is the skill of translating ordinary statements into mathematical ones

The list below are the word indicators for the following mathematical symbols:

Mathematical Symbols	Word Indicator
Equality	Is, is the same as, is equal to, equals, gives, yields
Addition	The sum, added to, increased, more than
Subtraction	The difference, substracted to, decreased, less than
Multiplication	The product, twice, doubled, tripled
Division	The quotient, divided by

The first digit is seven less than the second digit

The first step in translating this statement into a mathematical statement is to first identify the variables and denote them by letters. Consider

F = first digit

S= second digit

S – 7 = f

The length of the rectangle is twice its width. If the perimeter of the rectangle is 12, what is the width of the rectangle?

1.       Identify the variables

2.       Identify which of the variables is asked

3.       Translate the whole number mathematically

4.       Solve the problem

L = length of the rectangle

W = width of the rectangle000

L = 2W

P = sum of all sides a

  P = W + W + L + L