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Algebra

Mathematical Equation

Till now we spoke about numbers and shapes in Mathematics. However, the study of mathematics is divided into smaller branches. The study of numbers is Arithmetics. The study of shapes is Geometry. In this article let’s get introduced to another branch of mathematics – Algebra. This branch uses letters along with numbers to create Algebra formulas which helps us to solve complex problems with ease.

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The Matchstick Pattern Example

Rachna and Rohini are playing with some matchsticks and decide to make patterns using them. Rachna takes two matchsticks and makes the letter L. Then, Rohini picks up two sticks and forms another ‘L’ and puts is next to the one made by Rachna as shown below.

Algebra formulas

Eventually, Ram comes in and looks at what the girls are doing. Out of curiosity, he asks, “How many matchsticks will be needed to make 7 L’s in this manner?” Rachna and Rohini immediately start calculating and observe the following:

  • 1 L = 2 matchsticks
  • 2 L’s = 4 matchsticks
  • 3 L’s = 6 matchsticks
  • 4 L’s = 8 matchsticks.

Now, they start observing a pattern that the number of matchsticks required is twice the number of L’s formed. They turn to Ram and say, “You will need 14 matchsticks to make 7 L’s.” Ram is surprised by the immediate answer. He takes 14 matches and to his surprise form 7 L’s without having an extra stick.

What Rachna and Rohini did was simple. Without realizing they took the help of algebra formulas for the pattern as follows: the number of matchsticks required = 2n … where n = 1, 2, 3, 4, 5, etc. It is the number of L’s formed.

Note that the value of ‘n’ is not constant; it keeps changing and can take any value 1, 2, 3, 4 …. Hence, ‘n’ is known as a variable. You can use any alphabet in place of ‘n’ as a variable.

Use of Variables

Let’s see how we can use variables in some common rules in mathematics that we have already learned.

Geometry

  • The perimeter of a square: The perimeter of a polygon is the sum of lengths of all its sides. A square has four sides of equal length. Hence,

The perimeter of a square = Sum of lengths of all sides of the square
= 4 x length of a side of the square = 4l

Here l represents the length of a side of the square. Further, if we denote the perimeter as ‘p’, then we can write, p = 4l to denote the relationship between the length and perimeter of a square.

  • The perimeter of a rectangle: In a rectangle, the opposite sides are of equal length.

Algebra formulas

In the figure above, you can see that sides AB and CD are of equal lengths. Also, sides AD and BC have the same lengths. Let’s denote the lengths by letters as follows:

  • AB = CD = l
  • AD = BC = b

The perimeter of the rectangle = length of (AB + BC + AD + CD) = l + b + I + b = 2l + 2b
Hence, the perimeter of the rectangle p = 2l + 2b

Arithmetics

  • Commutativity of addition of two numbers: We know that, 4 + 3 = 3 + 4. We also know that this is true for any two numbers. This property is the Commutativity of addition of numbers. In simple words, changing the order of numbers, in addition, does not change the sum. We can express this rule as follows:

a + b = b + a      (where a and b can take any number values)

  • Commutativity of multiplication of two numbers: We know that 4 x 3 = 3 x 4. We also know that this is true for any two numbers. This property is the Commutativity of multiplication of numbers. In simple words, changing the order of numbers in multiplication does not change the product. We can express this rule as follows:

a x b = b x a     (where a and b can take any number values)

  1. Distributivity numbers: We know that, 7 x 38 = 7 x (30 + 8) = 7 x 30 + 7 x 8 = 210 + 56 = 266. We also know that this is true for any three numbers. This property is the Distributivity of multiplication over the addition of numbers. We can express this rule as follows:

a x (b + c) = a x b + a x c      (where a, b and c can take any number values)

Equations and Algebra formulas

Going back to the matchstick pattern example, we know that the number of matchsticks required to make a given number of L’s = 2n. Now, let’s say that Ram asks another question to Rohini and Rachna “How many L’s can be formed with 10 matchsticks?”

This changes the way we look at the expression. Now, we have, the number of L’s = ‘n’ and the number of matchsticks = 10. Hence,

2n = 10

So, we have a condition to be satisfied by a variable. This condition is an example of an equation.

Going by our earlier observations, we know that two matchsticks form one L, four form 2 L’s, six form 3 L’s, eight form 4 L’s and 10 form 5L’s. Hence, we find that the condition is satisfied only when n=5. For all other values of n, the condition (or equation) is not satisfied.

This value of the variable that satisfies an equation is known as the solution to the equation. So, the solution to the equation 2n = 10 is, n=5.

Solved Examples for You

Question: Ram, Rohini, and Rachna want to buy notebooks. The shopkeeper sells one notebook for Rs.20. Ram wants to buy 5 books, Rohini wants 7 books and Rachna wants 4 books. How much money should they carry?

Solution: To begin with,

  • The cost of one book = Rs.5
  • Cost of two books = Rs.10
  • Similarly, the cost of three books = Rs.15
  • Cost of 4 books = Rs.20 and so on.

They observe a pattern again and create the following rule: Total cost (in rupees) = 5m … where m is the number of notebooks required. Also, m = 1, 2, 3, 4, 5, etc. Now, Ram wants to buy 5 books. Hence, he needs to carry 5 x 5 = Rs.25. Rohini needs to carry 5 x 7 = Rs.35. Rachna needs to carry 5 x 4 = Rs.20.

Question: Complete the entries in the third column of the table:

Algebra formulas

Solution:

  1. 10y = 80. If y = 10, then 10y = 10 x 10 = 100 ≠ 80. Hence, the answer is NO.
  2. 10y = 80. If y = 8, then 10y = 10 x 8 = 80. Hence, the answer is YES.
  3. 10y = 80. If y = 5, then 10y = 10 x 5 = 50 ≠ 80. Hence, the answer is NO.
  4. 4l = 20. If l = 20, then 4l = 4 x 20 = 80 ≠ 20. Hence, the answer is NO.
  5. 4l = 20. If l = 80, then 4l = 4 x 80 = 320 ≠ 20. Hence, the answer is NO.
  6. 4l = 20. If l = 5, then 4l = 4 x 5 = 20. Hence, the answer is YES.
  7. b + 5 = 9. If b = 5, then b + 5 = 5 + 5 = 10 ≠ 9. Hence, the answer is NO.
  8. b + 5 = 9. If b = 9, then b + 5 = 9 + 5 = 14 ≠ 9. Hence, the answer is NO.
  9. b + 5 = 9. If b = 4, then b + 5 = 4 + 5 = 9. Hence, the answer is YES.
  10. h – 8 = 5. If h = 13, then h – 8 = 13 – 8 = 5. Hence, the answer is YES.
  11. h – 8 = 5. If h = 8, then h – 8 = 8 – 8 = 0 ≠ 5. Hence, the answer is NO.
  12. h – 8 = 5. If h = 0, then h – 8 = 0 – 8 = -8 ≠ 5. Hence, the answer is NO.
  13. p + 3 = 1. If p = 3, then p + 3 = 3 + 3 = 6 ≠ 1. Hence, the answer is NO.
  14. p + 3 = 1. If p = 1, then p + 3 = 1 + 3 = 4 ≠ 1. Hence, the answer is NO.
  15. p + 3 = 1. If p = 0, then p + 3 = 0+ 3 = 3 ≠ 1. Hence, the answer is NO.
  16. p + 3 = 1. If p = -1, then p + 3 = -1 + 3 = 2 ≠ 1. Hence, the answer is NO.
  17. p + 3 = 1. If p = -2, then p + 3 = -2 + 3 = 1. Hence, the answer is YES.

Question. What are the various rules of algebra?

Answer. The various rules and properties of algebra are as follows:

  • Commutative Property of the addition process.
  • Commutative Property of the multiplication process.
  • Associative Property of the addition process.
  • Associative Property of the Multiplication process.
  • Distributive Properties of Addition over the process of multiplication.
  • The reciprocal of a non-zero real number ‘a’ happens to be 1/a.
  • The additive inverse of a value ‘a’ is ‘-a’.

Question. What do we understand by ABC algebra?

Answer. ABC algebra deals with basic algebraic properties of real numbers a,b and c. These properties closure and commutative. For closure:  a + b and ab are real numbers. In contrast, for commutative: a + b = b + a, ab = ba.

Question. Who is known as the father of algebra?

Answer. Muhammad ibn Musa al-Khwarizmi is known as the father of algebra. He was a mathematician and astronomer in the 9th century.

Question. What do we understand by algebra?

Answer. Algebra refers to a branch of mathematics that deals with symbols. Furthermore, the subject also deals with the various rules for manipulating those symbols. In elementary algebra, those symbols are representative of quantities without fixed values.

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