Algebraic Expressions

Understand Factors and Coefficients of terms; Like and Unlike Terms, etc

Let us take some matchsticks and join them to form some patterns as shown in the following figure. The number of matchsticks used to make each pattern is written alongside it.

Now, let us try to represent the number of matchsticks used to form each pattern with the help of an algebraic pattern. The given pattern is formed by repeating the shape, which is made of 5 matchsticks.

Thus, we require 5 matchsticks to form the first shape, 8 matchsticks to form the second shape, 11 matchsticks to form the third shape, and so on. Thus, for each extra shape, we require 3 more matchsticks.

Now, we can represent the numbers 5, 8, 11, and 14 as expressions involving the numbers 1, 2, 3, and 4 as

5 = 3 × 1 + 2

8 = 3 × 2 + 2

11 = 3 × 3 + 2

14 = 3 × 4 + 2

Thus, we require (3n + 2) number of matchsticks to form n number of shapes.

We can verify this expression by taking n = 1, 2, 3 ….

If we take n = 3, then (3n + 2) = 3 × 3 + 2 = 9 + 2 = 11, which is indeed the number of line segments required to make the required shape.

Using this expression (3n + 2), we can calculate the number of line segments required to make any number of shapes. For example, if we want to make 25 such shapes, then we require (3n + 2) = 3 × 25 + 2 = 75 + 2 = 77 matchsticks.

If we actually draw the pattern and count the number of matchsticks, then it will be very time consuming. But the use of algebraic expression has made our job much simpler.

We come across these types of number patterns in Algebra.

Let us discuss one such number pattern, which does not involve any figure.

4, 13, 22, 31, 40 … 9n − 5

The term which occurs at the nth position in the pattern is given by the expression 9n − 5. If we want to find the number at the 50th position in the pattern, then we can easily calculate it by substituting n = 50 in this expression.

Thus, number at the 50th position in the pattern = 9 × 50 − 5 = 450 − 5 = 445

Similarly, the number in 91st position is 9 × 91 − 5 = 819 − 5 = 814

Let us discuss some more examples based on number patterns.

Example 1:

Observe the patterns of triangles made from matchsticks. Find the algebraic

expression that gives the number of matchsticks in terms of the number of triangles.

Find the number of matchsticks required to make 29 such triangles. Also find the number of triangles formed, if there are 51 matchsticks.

Solution:

The given pattern is formed by repeating the shape that is made from 3 matchsticks. The number of matchsticks required to form 1, 2, 3, 4 ... shapes are 3, 5, 7, 9 ... respectively. Here, we can see that for each extra shape, we require 2 more matchsticks.

3 = 2 × 1 + 1

5 = 2 × 2 + 1

7 = 2 × 3 + 1

9 = 2 × 4 + 1

In this way, we require (2n + 1) number of matchsticks to form n number of shapes.

To find the number of matchsticks required to form 29 such shapes, we require to substitute n = 29 in the expression, (2n + 1). Therefore, the number of matchsticks required to make 29 shapes is 2 × 29 + 1 = 58 + 1= 59.

To find the number of triangles formed from 51 matchsticks, we have to solve the equation 2n + 1 = 51.

2n = 51 − 1 = 50

Therefore, 25 triangles can be formed from 51 matchsticks.

Example 2:

Observe the number pattern 8, 13, 18, 23, 28 ………. 5n + 3.

Find the 31st and 67th terms of the pattern. Also find the place occupied by the term whose value is 253.

Solution:

It is given that the nth term of the expression is 5n + 3.

To find the 31st term, we have to substitute n = 31 in this expression.

Therefore, 31st term = 5 × 31 + 3 = 155 + 3 = 158

Similarly, to find the 67th term, we have to substitute n = 67.

Therefore, 67th term = 5 × 67 + 3 = 335 + 3 = 338

Now, to find the place occupied by the term whose value is 253, we have to find the value of n which satisfies 5n + 3 = 253.

5n = 253 − 3 = 250

Therefore, the 50th term of the pattern is 253.

Let us take some matchsticks and join them to form some patterns as shown in the following figure. The number of matchsticks used to make each pattern is written alongside it.

Now, let us try to represent the number of matchsticks used to form each pattern with the help of an algebraic pattern. The given pattern is formed by repeating the shape, which is made of 5 matchsticks.

Thus, we require 5 matchsticks to form the first shape, 8 matchsticks to form the second shape, 11 matchsticks to form the third shape, and so on. Thus, for each extra shape, we require 3 more matchsticks.

Now, we can represent the numbers 5, 8, 11, and 14 as expressions involving the numbers 1, 2, 3, and 4 as

5 = 3 × 1 + 2

8 = 3 × 2 + 2

11 = 3 × 3 + 2

14 = 3 × 4 + 2

Thus, we require (3n + 2) number of matchsticks to form n number of shapes.

We can verify this expression by taking n = 1, 2, 3 ….

If we take n = 3, then (3n + 2) = 3 × 3 + 2 = 9 + 2 = 11, which is indeed the number of line segments required to make the required shape.

Using this expression (3n + 2), we can calculate the number of line segments required to make any number of shapes. For example, if we want to make 25 such shapes, then we require (3n + 2) = 3 × 25 + 2 …