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Algebra

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Algebra is an important branch of Mathematics, which deals with several types of problems. These problems may either be related to constants or variables we mainly concentrate on "how to solve numbers.” When we solve any algebra problem there are several games using which you can easily learn the concept of algebra like kids have a great fun when they solve the puzzles, play some computer games by finding secret doors, etc.

When we deal with algebra, we mainly focus on equations and expression. These two terms can be defined as the heart of the algebra, as whenever we solve any problem we have to solve different equations.

Equations are mathematical statements, which show the equality of two different numbers or expressions.
Algebra is much broader than elementary algebra using algebra, we solve the equation problems. In algebra, we use different types of rules and operations, and perform all actions. In this, the variable symbol that represents numbers and expressions are mathematically termed as variables, numbers or both. On the different side, expressions are the mathematical phrases, which does not require equal to symbol as it does not show any equality.

Definition of Algebra

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Algebra is a vast branch of mathematics, where the relationship between two things that vary over the time are described using the mathematical statements. The mathematical statement that describes the relationship is called as algebraic expressions, terms or equations. Here, we use letters to represent the relationship, as the quantity varies and they are not a fixed amount. These letters are called as variables.

Elementary Algebra

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Elementary algebra deals with the basic concepts of algebra, which is one of the main concept of mathematics. It is a course for the students, who are starting from the beginning level of algebra.
Elementary algebra covers numbers, linear equations, polynomials, factoring, square roots, quadratic equations and graphing.
These topics also come under intermediate algebra or algebra 1. But, here in elementary algebra, we will study only the basic pace of those topics.


Given below are few problems based on elementary algebra topics.

Solved Examples

Question 1: Solve the linear equations given below:
7x - y = 5
5x - 10y = 21
Solution:
Given:
7x - y = 5 --> (1)
5x - 10y = 21 --> (2)

Solve equation (1) for y,
y = 7x - 5

Now, substitute the value of y in the equation (2)
5x - 10(7x - 5) = 21
5x - 70 + 50 = 21
- 65x = 21 - 50
- 65x = -29

x = $\frac{-29}{-65}$

= $\frac{29}{65}$

By subsitituting the x value in the equation y = 7x - 5, we get

y = 7($\frac{29}{65}$) - 5

y = $\frac{203}{65}$ - 5

y = $\frac{203 - 325}{65}$

y = $\frac{-122}{65}$

$\therefore$, x =  $\frac{29}{65}$ and y = $\frac{-122}{65}$

Question 2: Draw a graph for the equation y = 2x + 4
Solution:
Given: y = 2x + 4
At, x = 0
y = 2(0) + 4
y = 4

At, x = 1
y = 2(1) + 4
y = 6

At, x = - 2
y = 2(-2) + 4
y = 0

At, x = -3
y = 2(-3) + 4
y = -2

$\therefore$, co-ordinates are (0, 4), (1, 6), (-2, 0) and (-3, -2)


Elementary Algebra

Question 3: Solve the given quadratic equation x2 - 3x - 72 = 0, using the quadratic formula.
Solution:
Given: x2 - 3x - 72 = 0
We can solve this above equation using the quadratic formula: x = $\frac{-b \pm \sqrt{b^{2} - 4ac} }{2a}$

Here, a = 1, b = -3 and c = -72

Now, substitute the above values in the formula,

x(1, 2) = $\frac{-(-3)\pm\sqrt{(-3)^{2}-4(-72)} }{2}$

x(1, 2)  = $\frac{-(-3)\pm\sqrt{9-4(-72)} }{2}$

x(1, 2) = $\frac{3\pm \sqrt{279}}{2}$

$\therefore$ x1$\frac{3 + \sqrt{279}}{2}$ and x2$\frac{3 - \sqrt{279}}{2}$

Abstract algebra deals with advanced topics of algebra and these advanced topics deals with abstract algebraic structures. The important structures are groups, rings and fields. The important branches of abstract algebra are commutative algebra, representation theory and homological algebra. Discrete mathematics, elementary number theory, integers and linear algebra are sometimes considered as the part of abstract algebra.

Given below are few problems based on abstract algebra topics:

Solved Examples

Question 1: Find the 7th term for the following sequences:
8, 12, 16, .......
Solution:
Given: First term (a) = 8

Difference between next term and previous term, 12 - 8 = 4

and 16 - 12 = 4

$\therefore$ Common difference = 4

The general form of arithmetic progression is

$a_n$ = a + (n - 1)d

Now, $a_7$ = a + 6d

= 8 + 6 x 4

= 8 + 24

= 32

Therefore, the 7th term of the sequence is 32.

Question 2: Solve the following linear equations with fractions:
$\frac{5x}{3} - \frac{6}{7}$ = $\frac{x}{8}$
Solution:
Given: 
$\frac{5x}{3} - \frac{6}{7}$ = $\frac{x}{8}$

Multiply both sides by LCD of the denominator.

LCD of 3, 7, 8 = 168

168$(\frac{5x}{3}-\frac{6}{7}$ = $\frac{x}{8})$

After dividing and multiplying,  we get

280x - 144 = 21x

Subtract 21x both the sides

280x - 21x = 21x - 21x + 144

259x = 144

x = $\frac{259}{144}$

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