Chapter Overview

Suppose Aftab is some number of years old and Shabnam is always 3 years older.

Instead of writing a separate calculation for every possible age, we can use a letter.

Let Aftab's age be $a$.

Then Shabnam's age is:

This expression works for any value of $a$.

This is the main power of algebra: a single expression can describe many different situations.

1. Letter-Numbers

A letter-number does not always represent the same number.

For example, if:

then:

when $x = 3$,

and when $x = 10$,

2. Algebraic Expressions

Examples:

3. Writing Expressions from Situations

Algebraic expressions can describe real-life relationships.

Suppose one notebook costs ₹$p$.

The cost of 5 notebooks is:

If an additional pen costs ₹10, the total cost becomes:

4. Formulas

Letter-numbers allow us to write general mathematical rules.

For example, if each side of an equilateral triangle has length $s$, its perimeter is:

For a rectangle with length $l$ and breadth $b$:

5. Finding the Value of an Expression

To find the value of an algebraic expression, replace the letter-number by its given numerical value.

This is called substitution.

6. Multiplication Symbol in Algebra

In algebra, the multiplication sign is usually omitted.

Instead of writing:

we write:

Similarly:

and:

7. Terms in Algebraic Expressions

An expression can contain one or more terms.

Consider:

Its terms are:

• $5x$
• $3y$
• $-7$

For:

the terms are:

8. Like Terms

Examples of like terms:

They all contain $x$.

Similarly:

are like terms.

Examples of unlike terms:

9. Combining Like Terms

Like terms can be added or subtracted.

For example:

means:

Using the distributive property:

Therefore:

Similarly:

10. Simplifying Expressions

Simplifying means writing an expression in a shorter equivalent form.

Consider:

Combine the like terms:

Therefore:

11. Distributive Property

The distributive property helps us open brackets.

For example:

means:

Distribute 3:

Therefore:

12. Expressions and Number Patterns

Algebra helps us describe patterns without writing every term.

Consider:

These are multiples of 4.

The first term is:

The second is:

The third is:

Therefore the $n$th term is:

13. Matchstick Patterns

Suppose one square requires 4 matchsticks.

When another square is attached beside it, only 3 more matchsticks are required.

For $n$ connected squares:

Simplifying:

14. Calendar Patterns

Algebra can also explain patterns in calendars.

Suppose the top-left number in a $2 \times 2$ calendar block is $a$.

Then the block is:

| | | |---|---| | $a$ | $a+1$ | | $a+7$ | $a+8$ |

The first diagonal sum is:

The other diagonal sum is:

Therefore both diagonal sums are always equal.

Important Terms

Exam Focus

Quick Revision

Practice Questions

Multiple Choice Questions

Short Answer Questions

Numerical Questions

Apply Your Learning

Chapter Summary