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Negative Numbers

Multiplication of Integers

Just like on natural numbers, mathematical operations are also performed on algebraic expressions.

We can add and subtract like terms. For adding and subtracting like terms, only their numerical coefficients are added or subtracted; the algebraic part of the terms remains the same.

Consider the addition of two like terms, 8xy and 5xy.

The numerical coefficient of 8xy and 5xy are respectively 8 and 5; algebraic part of both the terms is xy.

Therefore, for adding 8xy and 5xy, we will add 8 and 5, keeping xy as it is.

∴ 8xy + 5xy = (8 + 5)xy = 13xy

Similarly, 5xy can be subtracted from 8xy in the following manner:

8xy – 5xy = (8 – 5)xy = 3xy

As by now we have learnt how to add and subtract like terms, let us learn how to simplify the algebraic expressions.

Let us first recall the two properties satisfied by all the numbers, x, y and z, which will be used to simplify the algebraic expressions.

Property I: x – (y + z) = (x – y) – z

Property II: x – (y – z) = x – y + z

Now, let us learn the method of simplification of expressions with the help of some examples.

Consider (2a + 5b) – (a + b).

For simplifying this, let us take x = (2a + 5b), y = a and z = b.

So, the given expression reduces to x – (y + z).

On using property I, we get:

x – (y + z) = (x – y) – z

i.e. (2a + 5b) – (a + b) = (2a + 5b – a) – b

= (2a – a + 5b) – b

= a + 5b – b

= a + 4b

Similarly, (3a + 2b) – (2a + b) can be simplified using property II, which is shown in the following manner:

(3a + 2b) – (2a + b)

= 3a + 2b – 2a – b [Here, x = (3a + 2b), y = 2a and z = b]

= 3a – 2a + 2b – b

= a + b

Value of an expression at a point:

Length and breadth of a rectangle are (3x + 2y) units and (2x – y) units respectively. Express the perimeter of the rectangle with the help of an algebraic expression.

We know that perimeter of a rectangle = 2 (Length + Breadth)

= 2 [(3x + 2y) + (2x – y)] units

= 2 [3x + 2y + 2x – y] units

= 2 [(3x + 2x) + (2y – y)] units

= 2 (5x + y) units    

= (10x + 2y)units

If we put x = 1 and y = 1, then the length of the rectangle = [3 (1) + 2 (1)] units = 5 units.

Breadth of the rectangle = [2 (1) – 1] units = 1 unit

Also, perimeter of the rectangle = [10 (1) + 2 (1)] units = 12 units

Similarly, if we put x = 2 and y = 1, then the length of the rectangle = 8 units.

Breadth of the rectangle = 3 units

Perimeter of the rectangle = 22 units

As seen here, the perimeter of the rectangle changes with the change in its length and breadth. In this way, we can find the value of an expression, at a given point, by substituting the values of the variables.

The value of an algebraic expression depends upon the value of its variables.

Using this concept, some of the properties of numbers can be verified.

Let us verify that the same for all positive numbers a, b, (a – b) ≠ (b – a).

In fact, (a – b) = –(b – a).

This can be verified in the following manner:

A

b

a – b

b– a

–(b – a)

1

2

–1

1

–1

1

–2

3

–3

3

–1

2

–3

3

–3

–1

–2

1

–1

1

2

1

1

–1

1

2

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