Introduction to Variables
Variables are generally used to assign some value; it is a symbol which represents a number. In general n, s, x are used as variables. We generally use these variables in formulas, and while determining unknown values. There are other type of variables called free and bond variables; Free variables are generally used variables and where as bond variables are correlated with each other.
- Free variables (Indepedent variable): n, x, s etc.
- Bond variables(Depedent variable): (x+2)2 = x2+4x+4 (or) (x2-4) = (x+2) (x-2)
Here in the above cases the variable x is related to both the sides.
Properties of Variables
1) Any equation can be added, subtracted or divided to get the variable value.
2) Variables play an important role in algebra and functions.
Types of Variables
Variables are used in equations depending on variables equations are divided into two types
1) Simple equations
2) Complex equations
3) Multiple variables
Simple Equations
These are basic equation which contain only one variable i.e. in left hand side or right hand side. Simple equations include whole numbers and one variable.
General form of equation is,
X+a=b
where,
X = variable
A and b are constants
Variable can be identified by different mathematical operations such as
1) Addition
2) Subtraction
3) Multiplication
4) Division etc
Complex Equations
Complex equations contain variable both in left hand side and right hand side. Equations contain fractions, decimals and multiple operations.
General form of equation is,
X+1=2x
Where,
X = variable
Here variable is present on both side i.e. left and right hand side.
Variables can be identified by different mathematical operations.
Multiple Variables
Multiple variables contain two or more variables on two sides i.e. left hand side and right hand side, we can identify one variable by using others.
General form of equation is:
X+Y=C
Where,
X, Y = variables
C = constant.
Mathematical Expression
Mathematical expressions are the combinations of mathematical symbols or numbers or both. Every expression contains important elements called variables. Expression generally does not contain equality sign, if it has an equality sign then it represented as an equation.
Example of an Expression
x2-2x+5 it is an expression
y=x2-2x+5 is an equation.
Properties of an Expression
• An expression should have a variables, symbols, and numbers.
• An expression should not have equality symbol.
Defined and Undefined forms
Defined and undefined forms these are very important things to be noted in expressions. They are generally used to give an expression meaningful value. The meaning of any expression depends up on the elements which are present in the expression i.e. symbols, variables, integers etc.
Example:
$\frac{0}{1}$, ∞, -∞, $\frac{0}{1}$, (1) × ∞, 0×0 etc...
Expressions can be obtained by different mathematical operations such as
1) Additions
2) Subtractions
3) Multiplication
4) Division
5) Factoring
Simplifying Expression
Algebraic expressions contain alphabetic symbols and numbers. When an algebraic expression is simplified, an equivalent expression is found more simpler than the original. This usually means that the simplified expression is smaller than the original expression.
Example of Simplifying Expression
Simplify x + y+2x-3y=0
Solution: Given expression,
x + y+2x-3y=0
Add and subtract the like terms we get ,
X+2x+y-3y=0
3x-y=0
Evaluating an expression with variables
Expression can be evaluated based on number of variables.
Single variable:
Example:
Evaluate the expression 9+x-3 if x=4
Solution: Given expression,
9+x-3
X = 4
Substitute x=3
9+4-3=9+1
= 10.
Two variable:
A mathematical expression contain variable as part of the expression. The variable with expression can be simplified by substituting the variables in expression.
Example:
Evaluate the expression 9+y+x+2 if x=2, y=1
Solution : Given expression,
9+y+x+2
x=2
y=1
9+2+1+2=14.
Variables and Expression Solved Examples
In case of finding area of a square we take the formula as s x s or s2. In this ‘s’ is the side of a square. We use the variable s instead of using side of a square.
1) Find the area of a square whose side is 4 cm?
Solution: Given that,
Side ‘s’=4
Area of a square = s2 = (4)2 = 16sq.cms
2) Jane has bought 5 apples and 6 oranges for 30, Find the cost of apples if each orange is $2 ?
Solution: We can write as,
5(A) +6(O) =$30
Each apple = $2
So we get 5(2) + 6(O) = $30
10 + 6(0) = 30
6(O) = 30-10
6(O) = 20
O = 20/6=3.3
Therefore each orange is $3.3.
In the above problem the question was changed into mathematical format using equations.
3) Find out the roots for the given quadratic expression x2-5x+6.
Solution :Given that
Quadratic expression is x2-5x+6
Make it into equation x2-5x+6=0
x2-3x-2x+6=0
x(x-3)-2(x-3)=0
(x-3) (x-2)=0
The roots are x= 3,2.
4) Plot the graph for the given expression y=x+3?
In the above graph each point is plotted by the expression y=x+3.
5) Evaluate the expression (1 +p) × 2 + 12 ÷ 3 - p when p= 3?
Solution :Given data,
Expression: (1 +p) × 2 + 12 ÷ 3 – p
p=3
We replace p with the number 3, and simplify using the usual rules;The order is simplify first parentheses , exponents, multiplication and division, addition and subtraction.
(1 + p) × 2 + 12 ÷ 3 - p
(1 +3) × 2 + 12 ÷ 3 - 3
4 × 2 + 12 ÷ 3 - 3
8 + 4 - 3
8+1=9.
6) Solve x - 12 + 20 = 37?
Solution :Given,
x-12+20=37
We need to find x values so “x” term should be independent, Hence we will bring constants to other side.
x-12+20=37
Bringing constants to other side
x=37+12-20
x=49-20
x=29.
7) John weighs 70 kilograms, and Mark weighs “ s ” kilograms. Write an expression for their combined weight?
Solution :Given data,
John weighs 70 kilograms
Marks weigh “ s ” kilograms
Expression for combined weight=?
The combined weight in kilograms of these two people is the sum of their weights, which is "70+s".