Algebra

You may have heard that Algebra is a difficult topic. Don't worry - here's a basic algebra lesson using a really simple way to get started. If you already know simple arithmetic, you are ready to start. 😄


What is Algebra ? 😕

As per Wikipedia

- Algebra is a branch of mathematics that deals with relations, operations and their constructions. It is one of building blocks of mathematics and it finds a huge variety of applications in our day-to-day life. 

Apart from its significance as a core subject of mathematics, Algebra helps students and kids a lot in developing an overall understanding of other advanced branches of mathematics such as Calculus, Geometry, Arithmetic etc. 

Advanced Definition of Algebra

Algebra (from Arabic al-jebr meaning "reunion of broken parts") is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. 

Concepts Associated with Algebra

😃 Elementary Algebra involves simple rules and operations on numbers such as:

•  Addition
•  Subtraction
•  Multiplication
•  Division
•  Equation solving techniques
•  Variables
•  Functions
•  Polynomials
•  Algebraic Expressions

There is no limit on the complexity of all these concepts as far as “Algebra” is concerned. Newer concepts and ideas may be added as the level of education rises. All these concepts are very useful and easy-to-understand if taught properly. 

The Basics

The first thing to grasp is that when we have an equation, both sides have exactly the same value. 

Let's start with:

6 = 6

That is an equation. Simple enough? Now we change the equation a little by introducing simple arithmetic operations that you already know:

4 + 2 = 6
6 = 2 × 3

Thus: 4 + 2 = 2 × 3

Easy to follow so far? OK, the next step is something you may done in arithmetic quizzes in grade school:

4 + = 2 × 3 

If you are asked to fill in the box, you can do the simple arithmetic and know that the answer should be 3. Now we are ready for basic algebra. Let's substitute the box with the letter 'k' and we have:

4 + k = 2 × 3

In the equation above, the letter 'k' is known as a variable. Of course we know that it is 3, so why is it called a variable? Well, that's the way algebra is - there are just some terms where the meaning is not as straightforward. 

You may think of it this way - if you were just given the equation 4 + k = 2 × 3 without any of the earlier discussions, then k would be unknown until you solve the arithmetic. 

That's the idea for variables in algebra. Anyway, variables are defined as numbers that can change value or represent a missing value (an unknown value). Variables are usually represented by letters of the alphabet, and the letters x, y, and z are most commonly used. 

Now we have a real basic algebra equation, and the goal is to solve for the variable k - that means to find the value of 'k' in the equation. Of course we know from earlier our earlier exercises that k = 3, but hey, where's the fun if algebra is just like that?

So, an algebra equation would be given as: 4 + k = 2 × 3 without any of the earlier exercises and you would be asked to solve for the unknown k.

The Fundamental Principle of Equation

Before we go about solving for the variable k, there's just one simple principle of equations that we need to grasp. Since we know that both sides of the equation are the same, whatever we do on one side (arithmetically), if we do the same to the other side, and the result is still an equation - that means both sides would still be equal.

 For example, we can do any of these:

4 + k - 2 = 2 × 3 - 2

4 + k + 4 = 2 × 3 + 4

(4 + k) × 3 = (2 × 3) × 3

Solving Our First Equation

Now we are ready to tackle our first algebra equation. What we want to do is to isolate the variable k on one side of the equation. Let's start with the equation:

4 + k = 2 × 3 

We can see that on the left side, there's an extra 4 added to k. So we must get rid of the 4 to isolate k. We can do this be subtracting 4 from the left side. Remember that we must do the same thing to the right side to maintain equality:

4 + k - 4 = 2 × 3 - 4

Now we are almost done solving our first algebra equation! Looking at the left side 4 + k - 4, the two 4s (4 and -4) would cancel out, leaving us with:

k = 2 × 3 - 4 

So we only need to do the arithmetic on the right side:

k = 2 × 3 - 4

k = 6 - 4

k = 2

With this you have a good understanding of basic algebra. 😃


😄 If it difficult to understand, this will make you remember it easily :

Example :

Expand the following :

a) 2 ( 𝚡 + 1)                         d) -3 ( 2b - 7)
b) 4 ( 3𝚡 - 5)                        e) 3𝚡 - 10 = 11
c) -2 ( a + 5)


Solution


a) 2 ( 𝚡 + 1) = 2 x 𝚡 +2 x 1                                    d) -3 (2b - 7) = -3 x 2b -3 x (-7) 
                    = 2𝚡 + 2                                                                  = -6b + 21


b) 4 ( 3𝚡 - 5) = 4 x 3𝚡 + 4 x (-5)                             e) 3𝚡 - 10 = 11
                     = 12𝚡 - 20                                                     3𝚡 = 11 + 10
                                                                                           3𝚡 = 21 
                                                                                             𝚡 = 21
c) -2 (a + 5) = -2 x a - 2 x 5                                                         3
                    = -2a - 10                                                          𝚡 = 7



Exercise : 

a) 2 (3a + 4)                                c) 4 (2a - 3)
b) 2a (3a + 4)                              d) 4 (c - 4)


Solution

a) 2 (3a + 4) = 2 x 3a + 2 x 4                                     c) 4 (2a - 3) = 4 x 2a - 4 x 3

                     = 6a + 8                                                                    = 8a - 12


b) 2a (3a + 4) = 2a x 3a + 2a x 4                                d) 4 (c - 4) = 4 x c - 4 x 4
                       = 6a² + 8a                                                              = 4c