Algebra

Linear equation Introduction Geometry   Order of Operation 

Introduction to Trig-identities Calculus Statistics and Probability 

What is Algebra?

In previous articles we discussed about Trig Identities.Today we are going to discuss about important part of Mathematics which is Algebra.Algebra is the part of mathematics that, together with geometry, numbers theory, and analysis. Algebra is the branch of mathematics that “which deals with symbols and the rules for solving these symbols.”

Algebra introduces variable. In Arithmetic poses questions like 5+10=? But Algebra poses questions like x +10=20, find the value of x? Instead of solving through basic arithmetic, we can solve by using Algebra rules.

Example 

 a– 20 =40

Find the value of a?

Subtracting both sides by 20

           X-20 + 20 =40 + 20

           X = 60

Why we use these letters in Algebra?

We use these letters in algebra because it’s quite easy than drawing empty boxes and easy to write “X” instead of writing “Empty Boxes.”

If the equation is greater than two symbols than we use different letters for each empty space.

Adding or subtracting in Algebra?

In Algebra, if you adds or subtracts any terms, they must be of the same symbol name. Different symbol name variables can’t be added or subtracted.

Adding Example

                  4X + 10Y + 15X + 20Y

                  (4X +15X) + (10Y + 20Y)

                  (4 + 15) X + (10 + 20) Y

          Answer: 19 X + 30 Y

Subtracting Example 

                  – 8X – 5Y – 2X -3Y

                  (-8X – 2X) – (-5Y – 3Y)

                  (- 8 – 2)X – (-5 – 3) Y

                  (- 10)X – (- 8) Y

Answer       -10X -8 Y

Example

         10X + 2Y -3X -1Y

         (10X -3X) + (2Y – 1y)

         (7X) + (1y)

Answer: 7X + Y

Multiplying and Dividing Algebra equations:

In Algebra multiplication or division, you don’t have like terms. That’s the basic difference between addition and subtraction. Be careful while multiplying or dividing. Take a look at these below examples:

Multiplying Example

                       (X + 10) * 2

                  (X * 2) + (10 * 2)

Answer: 2X + 20.

 In the above equation, we multiply every part of the equation by 2.

Take a look at another example. Then your concepts look much better.

                       (X + 5) * 4

                  (X * 10) + (5 * 4)

                  (10X) + (20)

Answer: 10X + 20

Dividing Example

           5 X =30

Dividing both sides by 5

           5X / 5 = 30 / 5

         X= 6

Example

         X / 2 = 10

Multiplying both sides by 2

2 * (X / 2) = (10) * 2

X = 20

Properties of Algebra

Algebra holds the basic properties of real numbers:

  • Commutative property of addition
  • Commutative Property of Multiplication
  • Associative property of Addition & Multiplication
  • Distributive Property

Commutative property of addition

The word “commutative” comes from “Commute,” which means “Move around,” so the commutative property is that if we change the order of values result can’t change. Look at the example below.

Example

           A + B = B + A

         20 + 5 = 5 + 20

Algebraic Expression

         X2 + X = X + X2

         (2)2 + (2) = (2) + (2)2

         4 + 2 = 2 + 4

Answer is always same.

Commutative Property of Multiplication

The word “commutative” comes from “Commute,” which means “Move around,” so the commutative property is that if we change the order of values product can’t change. Look at the example below.

Example

                       (X * Y) = (Y * X)

                       (4 * 3) = (3 * 4)

           Answer 12 = 12

Algebraic Expression

         (X2 + 3X)(2X + 2) where x belongs to R.

(X2 + 3X)(2X + 2) = (2X + 2)(X2 + 3X)

Suppose X= 2

(22 + 3(2))(2(2) + 2) = (2(2) + 2) (22 + 3(2))

(4 + 6)(4 + 2)= (4 + 6)(4 + 2)

10 * 6 = 10 * 6

Answer 16 = 16

Associative property of Addition & Multiplication

Associative property defines that grouping of two numbers or more than two numbers performing basic mathematic operations don’t affect the result.

Example

Consider the following expression 2x + (x + y)for addition where x & y belong to R.

2x + (X + Y)= (X + Y) + 2X

Let suppose x =1 and y=2

2(1) + (1+ 2) = (1 + 2) + 2(1)

2+ 3 = 3 + 2

Algebraic expression ((X + 2) * 3X2) * 4x = (x+2) * (3X2 * 4X) holds the property for multiplication.

Distributive Property

The distributive property defines that the multiplication of a single value and a sum or difference of two or more values inside the bracket is same as multiplying each addend by the single term and then adding or subtracting the products.

a *(b + c) = (a * b)(a* c)

4X(X2 + X4) = 4X(X2) + 4X (X4) = 4X3 + 4X5

4X(X2 – X4) = 4X(X2) – 4X (X4) = 4X3- 4X5