Algebra is the language of generalization or simply claim as simplification. If you do a work once then, you probably don’t need algebra. But if a work is done repeatedly, algebra provides a single language describing the work done. Algebra enables a person to answer all the questions of a particular type at one time. Algebra is the language of relationships between quantities. Algebra solves numerical problems.

Algebra is a language that describes patters, for example

We all know,

Hence,

$\frac{3}{2}$ $\times$ $\frac{5}{4}$ = $\frac{15}{8}$

This when written in the algebraic language:

$\frac{a}{b}$ $\times$ $\frac{c}{d}$ = $\frac{ac}{bd}$

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- Letters signify an unknown or generic real number
- Sometimes with limitations, such as a member of a certain set, or the set of values that makes an equation true
- Often a letter from the end of the alphabet: x, y, z
- Or a letter that stands for a physical quantity: d for distance, t for time, etc.

- Fixed values, like 2, 5 or 7.
- Can also be represented by letters: a, b, c, p , s, r

Terms are Separated by + or –

2 x - x + 3 Terms

Factors are multiplied together.

By “simplifying” an algebraic expression, we mean writing it in the most compacted or efficient manner, without changing the value of the expression. This mostly comprises collecting like terms, which means that we add together anything that can be added together. The rule is simple that only like terms can be added together. **Like terms** are those terms which contain the same powers of same variables. They can have different coefficients, but that is the only difference.

- $5x$, $x$, and –$2x$ are like terms.
- $4x^2$ , –$7x^2$ , and $2^2$ $1x$ are like terms.
- $xy^2$, $3y^2x$, and $3xy^2$ are like terms.
- $xy^2$ and $x^2y$ are NOT like terms, because the same variable is not raised to the same power

There are certain strategies in algebraic problem solving such as:

- Read the algebraic problem attentively.
- Ensure you understand the situation that is being defined.
- Learn what information is provided, and what the question is asking.
- For most of the problems, drawing a clearly classified picture is very advantageous.

- Target on the objective.
- Check the given information. How the information can be used to answer the question?
- If you do not see a definite logical path leading from the given information to the solution.

You need to prompt mathematically the logical connections between the given information and the answer you are in search of. This involves:

1. **Conveying variable names to the unknown quantities: **The letter x is always popular, but it is a good idea to use something that reminds you what it represents, such as d for distance or t for time. If you know how two numbers are related, then you can prompt them both with just one variable. For example, if Jack is two years older than Joan is, you might let x stand for Joan’s age and

(x + 2) stand for Jack’s age.

2. **Translate English into Math:** Mathematics is a language, that is apt to describe reasonable relations.

- Solve - Solve the equation(s) for the unknown(s).
- Check
- Think - Does your answer come out in the correct entities? Is it reasonable?

How can we do maths if there are no symbols to express, for instance, the consecutive general fact about a positive integer n?

The difference of any two nth powers is equal to the product of the difference of the two numbers and the sum of products comprising of the (n − 1)^{th} power of the first number, then the product of the (n − 2)^{th} power of the first and the first power of the second, then the product of the (n−3)^{th} power of the first and the second power of the second, and so on, until the (n−1)^{th} power of the second number.

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