**What is Algebra**

**Algebra** is derived from Arabic word al-jebr meaning “reunion of broken parts”. Algebra 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.

Algebra is also considered as a mathematical segment that exchanges letters for the number. An algebraic equation is working as a scale where both the sides signify the same things as well as numbers are taken as the constants. You may check the complete meaning of Algebra, Definition, Formula, Problems, Basics, Solved Examples etc from here.

__Algebraic Terms__:

The basic unit of an algebraic expression is a term. In general, a term is either a number or a product of a number and one or more variables. Below is the term – 3ax.

*Numerical Coefficient=-3 and Variables= ax*

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**Algebra**

__Definition of 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.

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__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 and Algebraic Expressions

__Algebra Formulas Include Topics__:

- What is Algebra?
- List of Basic Algebra Formulas
- Basic Algebra Equations
- Algebra Problems with a solution
- Why Algebra Formula Needs etc.?

__List of Basics Algebra Formulas__:

- a2 – b2 = (a – b)(a + b)
- (a+b)2 = a2 + 2ab + b2
- a2 + b2 = (a – b)2 + 2ab
- (a – b)2 = a2 – 2b + b2
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
- (a – b – c)2 = a2 + b2 + c2 – 2ab – 2ac + 2bc
- (a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)
- (a – b)3 = a3 – 3a2b + 3ab2 – b3
- a3 – b3 = (a – b)(a2 + ab + b2)
- a3 + b3 = (a + b)(a2 – ab + b2)
- (a + b)3 = a3 + 3a2b + 3ab2 + b3
- (a – b)3 = a3 – 3a2b + 3ab2 – b3
- (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4)
- (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4)
- a4 – b4 = (a – b)(a + b)(a2 + b2)
- a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)
- If n is a natural number, an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)
- If n is even (n = 2k), an + bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)
- If n is odd (n = 2k + 1), an + bn = (a + b)(an-1 – an-2b +…- bn-2a + bn-1)
- (a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + ac + bc + ….
- Laws of Exponents
- (am)(an) = am+n
- (ab)m = ambm
- (am)n = amn
- Fractional Exponents
- a0 = 1
- aman=am−naman=am−n
- amam = 1a−m1a−m
- a−ma−m = 1am

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__Basic Algebra Equations__:

In Algebra, a basic equation is an expression that contains minimum one variable that we need to calculate to find new one which is unknown to you. For Example – (x – 38 = 12). This is an algebraic equation where we need to calculate the unknown variable x. We can calculate as follows:

For Example: (x – 38 = 12).

X=12+38

X=50

**Question 1:** Find out the value of 52 – 32

**Solution**:

Using the formula a2 – b2 = (a – b)(a + b)

where a = 5 and b = 3

(a – b)(a + b)

= (5 – 3)(5 + 3)

= 2 ×× 8

= 16

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**Question 2**: 43 ×× 42 = ?

**Solution**:

Using the exponential formula (am)(an) = am+n

where a = 4

43 ×× 42

= 43+2

= 45

= 1024

**Question 3**: 5(-3x – 2) – (x – 3) = -4(4x + 5) + 13

**Solution:** Multiply factors.

-15x – 10 – x + 3 = -16x – 20 +13

Group like terms.

-16x – 7 = -16x – 7

Add 16x + 7 to both sides and write the equation as follows

0 = 0

The above statement is true for all values of x and therefore all real numbers are solutions to the given equation.

**Question 4**: 2(a -3) + 4b – 2(a -b -3) + 5

**Solution:** Multiply factors.

= 2a – 6 + 4b -2a + 2b + 6 + 5

Group like terms.

= 6b + 5

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**Question 5**: |x – 2| – 4|-6|

**Solution: **If x < 2 then x – 2 < 0 and if x – 2 < 0 then |x – 2| = -(x – 2).

Substitute |x – 2| by -(x – 2) and |-6| by 6 .

|x – 2| – 4|-6| = – (x – 2) – 4(6) = – x -22

**Question 6**: (-4 , -5) and (-1 , -1) is given by

**Solution: **d = √[ (-1 – (-4)) 2 + (-1 – (-5)) 2 ]

Simplify.

d = √(9 + 16) = 5

**Question 7**: 2x – 4y = 9

**Solution:**To find the x intercept we set y = 0 and solve for x.

2x – 0 = 9

Solve for x.

x = 9 / 2

The x intercept is at the point (9/2 , 0).

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**Question 8:** Solve the following system of equations

– x / 2 + y / 3 = 0

x + 6 y = 16

**Solution: **We first multiply all terms of the first equation by the LCM of 2 and 3 which is 6.

6(-x/2 + y/3) = 6(0)

x + 6y = 16

We then solve the following equivalent system of equations.

-3x + 2y = 0

x + 6y = 16

which gives the solution

x = 8/5 and y = 12/5

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**Question 9:** Solve the following quadratic equation.

0.01 x 2 – 0.1 x – 0.3 = 0

**Solutions:** Multiply all terms of the equation by 100 to obtain an equivalent equation with integer coefficients. x 2 – 10 x – 30 = 0

Discrminant = (-10) 2 – 4(1)(-30) = 220

Solutions x = (10 ~+mn~ 2√55) / 2 = 5 ~+mn~ √55

**Question 10:** In a cafeteria, 3 coffees and 4 donuts cost $10.05. In the same cafeteria, 5 coffees and 7 donuts cost $17.15. How much do you have to pay for 4 coffees and 6 donuts?

**Solutions:**Let x be the price of 1 coffee and y be the price of 1 donut.

We now use “3 coffees and 4 donuts cost $10.05” to write the equation

3x + 4y = 10.05

and use “5 coffees and 7 donuts costs $17.15 ” to write the equation

5x + 7y = 17.15

Subtract the terms of the first equation from the terms of the second equation to obtain

2x + 3y = 7.10

Mutliply all terms of the last equation to obtain

4x + 6y = 14.2

4 coffees and 6 donuts cost $14.2.

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__Final Words__:

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