ICSE Class 6 Maths Chapter 12 Fundamental Concepts

Unit 3 - Algebra

Chapter 12 - Fundamental Concepts

12.1 Algebra

Algebra is a generalized form of arithmetic. In arithmetic, we use numbers like 5, -8, 0.64, etc., each with a definite value, whereas in algebra, we use letters (a, b, c, ..., x, y, z, etc.) along with numbers.

For example: 7x, 3x - 2, 5a + b, 2y - 5x, x + 2y - 7z and so on

The letters used in algebra are called variables or literal numbers or simply literals. They do not have a fixed value.

12.2 Signs And Symbols

In algebra, the signs +, -, x and ÷ are used in the same sense as they are used in arithmetic.

Also, the following signs and symbols are frequently used in algebra, each with the same meaning in every branch of mathematics.

12.3 Writing A Given Statement In Algebraic Form

Conversely

Teacher's Note

When you calculate the total cost of items at a store by multiplying price and quantity, you are using algebra to generalize a real-world transaction.

Exercise 12 (A)

1. Express each of the following statements in algebraic form:

(i) The sum of 8 and x is equal to y.

(ii) x decreased by 5 is equal to y.

(iii) The sum of 2 and x is greater than y.

(iv) The sum of x and y is less than 24.

(v) 15 multiplied by m gives 3n.

(vi) Product of 8 and y is equal to 3x.

(vii) 30 divided by b is equal to p.

(viii) z decreased by 3x is equal to y.

(ix) 12 times of x is equal to 5z.

(x) 12 times of x is greater than 5z.

(xi) 12 times of x is less than 5z.

(xii) 3z subtracted from 45 is equal to y.

(xiii) 8x divided by y is equal to 2z.

(xiv) 7y subtracted from 5x gives 8z.

(xv) 7y decreased by 5x gives 8z.

2. For each of the following algebraic expressions, write a suitable statement in words:

(i) 3x + 8 = 15

(ii) 7 - y > x

(iii) 2y - x < 12

(iv) 5 ÷ z = 5

(v) a + 2b ≥ 18

(vi) 2x - 3y = 16

(vii) 3a - 4b > 14

(viii) b + 7a < 21

(ix) (16 + 2a) - x > 25

(x) (3x + 12) - y < 3a

12.4 Constants And Variables

In algebra, we come across two types of symbols, namely, constants and variables.

A symbol with a fixed numerical value in all situations is called a constant, e.g. 5, 30, 256, -7, \(\frac{5}{3}\), \(\frac{7}{9}\), etc., whereas a symbol whose value changes with situation is called a variable, such as; x, y, p, q, 5x, etc.

In 3x, 3 is a constant and x a variable but, together, 3x is a variable. Reason: As the value of x will change, the value of 3x will also change accordingly. Similarly 3 is constant and x is variable but, together, each of 3 + x, x - 3 and x ÷ 3 is a variable. So, we conclude that every combination of a constant and a variable is always a variable.

Teacher's Note

The price of a single item is constant, but the total cost of multiple items is a variable that depends on how many you buy.

12.5 Term

A term is a constant or a variable or a product or a quotient of constants and variables.

For example:

(i) 4 is a term; which is a constant

(ii) x is a term, which is a variable

(iii) 4x is a term; which is the product of a constant and a variable.

(iv) \(\frac{3}{x}\) is a term; which is the quotient of a constant and a variable.

A term is called a constant term if it does not contain any literal (variable). Thus, each of 3, -20, \(\frac{5}{7}\), \(\frac{4}{9}\), etc. is a constant term.

Constants (fixed numbers) and variables (literal numbers) may be combined to form several types of terms.

For example:

The constants 2, 5, -8, 4, \(\frac{3}{2}\), etc., and the variables x, y, z, etc., may be combined to form terms such as 2x, 5y, 5xy, 5xyz, 4xz, \(\frac{3}{2}\)yz, ...

(i) Like Terms

The terms having the same literal coefficients are called like terms. They may differ only in their numeral coefficients.

For example:

(i) xy, 5xy, - 4xy, etc. are like terms

(ii) - 8x²y, 7x²y, 1.5x²y, etc. are like terms

Each having the same literal coefficient: xy and so on.

(iii) Unlike Terms

The terms that do not have the same literal coefficients are called unlike terms.

For example:

(i) 6a, 6ab and 6ac are unlike terms.

(ii) 2xy, 2x²y and 2xy² are unlike terms and so on.

12.6 Algebraic Expressions

An algebraic expression is a collection of one or more terms, which are separated from each other by the signs + (plus) and/or - (minus).

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