An algebraic equation is an equation that includes one or more variables. Any expression that contains letters, numbers and algebraic operations, addition(+), subtraction(-), multiplication(×), division(÷) and exponentiation to rational exponent is called an

An algebraic expression may refer to constants or variables or the combinations of both with associated operators. Variables are used to represent unknown quantities. In another way we can define algebraic expression as an algebraic model of a verbal model of a statement.**Algebraic Expression**. Expression is a sentence of words, it may have a literal meaning.**In mathematics**it specifically refers to all possible constants, variables and associated operations or any of the combinations between them.To understand clearly what is an algebraic expression, let us see an example:

### Solved Example

**Question:**If the sum of two numbers is equal to 10. What are the numbers?

**Solution:**

let us assign the variable ‘x’ for one number

and variable ‘y’ for another number

Thus the part of the given verbal statement ‘sum of two numbers’ can algebraically be written as ’x + y’.

ie

On substituting value of x, we will get the value of y.

if x = 6 then 6 + y = 10, y = 10 - 6 = 4

So the values of x and y are 6, 4 respectively.

and variable ‘y’ for another number

Thus the part of the given verbal statement ‘sum of two numbers’ can algebraically be written as ’x + y’.

ie

**x + y = 10**On substituting value of x, we will get the value of y.

if x = 6 then 6 + y = 10, y = 10 - 6 = 4

So the values of x and y are 6, 4 respectively.

The following are a few examples that illustrate different types of expression.

**x + y**is an expression that involve an addition.

**x * y**is an expression involving a multiplication.

**a**

^{n }is called a power expression, if ‘a’ is a real number and n is a positive integer.

**a**is called as exponential expression because in this case the variable is at the place of exponent.

^{x}### Solved Example

**Question:**Algebraic expression ‘x – y’ means ‘the difference of two quantities’

**Solution:**

Suppose we say x = 10 and y = 4, then the expression transforms as ’10 – 4’, which is operational now.

The difference of 10 and 4 is 6

Hence the value of the expression ‘x – y’ is 6 when x = 10 and y = 4.

This process is called ‘

The difference of 10 and 4 is 6

Hence the value of the expression ‘x – y’ is 6 when x = 10 and y = 4.

This process is called ‘

**evaluation of algebraic expression**’.**For example,**(xy + x) can be simplified as x(y + 1) by use of distributive property.

Similarly ln (x) + ln (y) can be simplified as, ln (xy) by use of logarithmic property.

The simplification of algebraic expressions helps a lot in solving for the variables. Mathematicians framed a rule for knowing the correct order which is known as the order of operations. It is abbreviated as PEDMAS, to indicate the correct order for simplifying an algebraic expression.

- "P" indicates that first the terms inside the parentheses must be simplified.
- "E" indicates evaluate all the exponents.
- "D" indicates division.
- "M" indicates multiplication.
- "A" indicates addition.
- "S" indicates subtraction.

**Example**:

8 ÷ 4 * 2 must be simplified as (8 ÷ 4) * 2 = 2 * 2 = 4

and

**not as**8 ÷ (4 * 2) = 8 ÷ 8 = 1.

### Solved Example

**Question:**If the statement is ‘4 more than twice the number is same as 10 reduced by the same number’.

**Solution:**

Let us assume the number as

Similarly ‘

Since it says ‘is same as’, the two expressions are equal. That is, we can form an equation as,

Now with our knowledge in algebra we can simplify and solve as

**x**.**4 more than twice the number**is expressed as an algebraic expression as**2x + 4.**

Similarly ‘

**10 reduced by the same number’**in algebraic expression form is**10 – x.**

Since it says ‘is same as’, the two expressions are equal. That is, we can form an equation as,

**2x + 4 = 10 – x.**Now with our knowledge in algebra we can simplify and solve as

**x = 3**. Thus we find the unknown number as 3.