Elementary algebra is the study of the arithmetic operators (+, -, *, /) and their rules along with the combinations of algebraic variables. If we combine some constants with some unknown variable, together with some arithmetic operations, then the expression is termed as an algebraic equation. The following are some of equations, representing algebraic equations or algebraic expressions.

Consider the following algebraic expression:
 Algebraic Expression
In the above mentioned algebraic expression or polynomials, 5x², 2y, 9xy, and 4 are known as terms.
Consider the term 9xy, here xy is the variable and the coefficient is 9. Similarly, in the term 2y, 2 is the coefficient and y is the variable.
There can be 3 types of algebraic expressions. A monomial that contains only one term, a binomial expression that contains two unlike terms and a trinomial that contains exactly three unlike terms. By looking at the above mentioned figure of algebraic expression we can clearly differentiate between these.

Elementary algebra includes many topics, out of which the most commonly used are mentioned below:

  • Factorization of ax2 + bx + c = 0 (Quadratic equations) over the set of integers.
  • Operations with the algebraic fractions involving addition, subtraction, multiplication, and division.
  • Division of monomials and other polynomials including the simplification of algebraic fractions.
  • Solution to the linear equations and inequalities.
  • Solving the simultaneous system of linear equations.
  • Quadratic equations solution by the method of factoring.
  • The conversion of written phrases or sentences into algebraic expressions or equations.
  • Solving any verbal problem in an algebraic way including its geometrical reasoning through graphing.

There are in numerous formulas in elementary algebra because we can derive many formulas from a given single formula. However, the following is the list of some of the important formulas used in elementary algebra:

A)    Basic Laws of Elementary Algebra: Assuming that a and b are real numbers we have the following laws:
1.    Closure Law:  a + b and ab are real numbers
2.    Commutative Law: a + b = b + a, (for addition), ab = ba (for multiplication)
3.    Associative Law: (a + b) + c = a + (b + c) (for addition), (ab) c = a (bc) (for multiplication)
4.    Distributive Law: (a + b) c = ac + bc
5.    Identity Law: a + 0 = 0 + a = a
6.    Inverse Law: a + (-a) = 0, a ($\frac{1}{a}$) = 1, which means every number has its additive inverse.
7.    Cancellation Law: If a + x = a + y, then, we can cancel a from both the sides, to get x = y
8.    Zero-factor Law: a * 0 = 0 * a = 0, multiplication with zero always gives the resulting answer as zero.
9.    Negation Law: - (-a) = a, (-a) b = a(-b) = -(ab), (-a)(-b) = ab, negative of a negative number is a positive number and so on.

B)    Factorization Formulas: The following formulas are used to factorize an algebraic expression according to their powers, as mentioned below:
1.    ( a – b ) ( a ²  + a b + b² ) =  a ³ – b ³
2.    ( a – b ) ³  =  a ³  – 3 a² b + 3 a b²  – b ³
3.    ( a + b ) ( a ²  – a b + b ² ) =  a ³ + b ³
4.    ( a + b ) ³  =  a ³  + 3 a² b + 3 a b²  + b ³
5.    ( a + b ) ( a – b ) = a ²  –  b ²
6.    ( a – b ) ²  =  a ² – 2 a b + b ²
7.    ( a + b ) ²  =  a ²  + 2 a b + b ²

C)    Quadratic Equation formulas: A quadratic equation is an algebraic equation or expression written in the second degree, as shown below:
ax2 + bx + c = 0, where  a, b, c are the given numerical values of the coefficients, and x is an unknown variable. The roots of any quadratic equation can be found by any of the following methods:
1.    The quadratic formula:

 $x = $$\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$

2.    Factoring Method: Here we split the middle term bx in such a way that the sum of the roots should be (–$\frac{b}{a}$) and the product of the roots should be ($\frac{c}{a}$).
3.    Completing the Square Method: In this method we make a perfect square of c, by adding ($\frac{1}{2}$ of coefficient of x) ² to both sides of the equation, to get a perfect or complete square.

D)    Graphing of the quadratic equation: The discriminant of a quadratic function = b2 - 4ac, where,
•    If the discriminant is positive, that is greater than zero, then it will have two real roots and the graph will have two x-intercepts.
•    If the discriminant is equal to zero, then the two roots will be real and equal to one another, and the graph will have only one x-intercept.
•    If the discriminant is less than zero, then the two roots will be imaginary and the graph will have no x-intercepts, and
a.    The graph will be drawn completely above the x-axis if a > 0.
b.    The graph will be drawn completely below the x-axis if a < 0.
Elementary algebra is done in such a way so that we can represent by analyzing and generalizing a variety of patterns within the tables, their graphs, words, and, whenever possible, the symbolic rules.
The basic procedure to solve any algebraic equation is shown as below:

Division
ax = b

$\frac{a}{a}$$x = $$\frac{b}{a}$

x = $\frac{b}{a}$

Multiplication
$\frac{x}{a}$ = b

a$\frac{x}{a}$ = ab

x = ab

Subtraction
x + a = b

x + a - a = b - a

x = b - a

Addition
x - a = b

x - a + a = b + a

x = b + a

Inversion
$\frac{1}{x}$ = a

x = $\frac{1}{a}$

Squaring

$\sqrt{x}$ = a

x = $a^2$

Square Root
$x^2 = a$

$x = \pm \sqrt{a}$

Natural Log
$e^x = a$

In($e^x$) = In a

x = In a

Exponentials
In x = a

$e^{In\ x} = e^a$

x = $e^a$


We know that, a set of values of unknown variables satisfying each one of the equation in the given system of two simultaneous equations is termed as a solution of the linear equations. And we can adopt any of the following methods to solve a linear equation in two variables:
1)    Graphical Method: If (a1, b1) and (a2, b2) are any two solutions of the equation ax + by + c = 0, then on plotting these points on a graph paper and joining these points, we will obtain a line, called the graph of the equation ax + by + c = 0. Note that every point on the line gives its solution.
2)    Substitution Method: In this we substitute the value of one variable from one equation to other equation to get, the value of other variable.
3)    Elimination Method: In this, we eliminate one variable to make only single variable as unknown, and then substitute the value in any equation to get the value of other variable.
4)    Cross Multiplication Method: Here we apply special rule of cross multiplication to solve for two variables.

Example:
 How to do Elementary Algebra
Thus, by splitting the middle term, we can get the two roots of the equation easily.
The following are some practice problems of elementary algebra:
1)    Convert the following statements into an algebraic expression:
a)    ten less than thrice a number
b)    two-third of a number

2)    Using the commutative law of multiplication find an algebraic expression which is equivalent to:
    ab + c.

3)    Using the associative law of addition find an algebraic expression which is equivalent to:
    (ab + c ) + 3

4)    Simplify: 18x + 10 = 14x + 110


5) Find the slope of the given lines from the following given graphs:
 (a)                                                                                        
 Elementary Algebra Practice Problems

b)

Practice Elementary Algebra
Answer: a) $\frac{1}{2}$ b) undefined



There could be many more questions from our day to day life by which we can do a lot of practice of elementary algebra as most of the solutions require the knowledge of elementary algebra only.
Some of the other solved questions of elementary algebra are as follows:

Solved Examples

Question 1: Solve the following algebraic equations and by making a graph of the solution on a number line.
a.    x – 3 > 1
b.    2 – 4y $\geq $ 7
c.    9 + 2y $\leq $ 4y + 5
Solution:
 
a.        x – 3 > 1 this implies  x > 1 + 3 ,
Hence, x > 4, and the graph is as follows:
 Number Line
b.        2 – 4y $\geq $ 7, this implies, -4y $\geq $ 7 – 2,
Hence, y $\leq $ -5 / 4 (multiplying with a negative sign changes the inequality), and the graph of the solution on a number line is as follows:
Number Line Example
c.        9 + 2y $\leq $ 4y + 5, this implies, 2y – 4y $\leq $ 5 – 9,
Hence, y $\geq $ 2 and the graph of the solution is as follows:
Solving Number Line
 

Question 2: Consider the following linear equation:
5x + 3y = - 6, then
a.    Find the x and y intercept along with the slope.
b.    Graph the line of the above linear equation.
Solution:
 
a.    The x-intercept is [ ($\frac{-6}{5}$), 0], the y-intercept is (0, -2) and the slope = $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ = $\frac{-2-0}{0-\frac{-6}{5}}$ = $\frac{-5}{3}$

b.    The graph of the line is shown as below: (by plotting the points of x intercepts and y intercepts and then joining them)

Elementary Algebra Graph
 

Question 3: Look at the following inequality and then draw its graph.
20 x – 30 y > - 60
Solution:
 
In the above inequality, after dividing the inequality by 10, we will get the reduced inequality, in which the x intercept is (-3, 0) and the y intercept is (0, 2) and the inequality is greater than zero, hence we will shade the upper part of the line as shown below:
 Elementary Algebra Graphs
 

Question 4: Solve the following linear equations simultaneously by the method of substitution:
2 x – 3 y = –2
4x +   y = 24.
Solution:
 
In the method of substitution, we will have to solve any one equation from the given two, to find the value of one variable, and then we will putt that value in the other second equation.
Consider 4x + y = 24, it can be rewritten as:
y = – 4x + 24, Substituting the value of "y" in the first equation, and solving for x, we will get, 2x – 3(–4x + 24) = –2 or 2x + 12x – 72 = –2 or 14x = 70, this implies:
x = 5. Now putting this value in the other equation, we will get,
y = –4(5) + 24 = –20 + 24 = 4
Hence, (x, y) = (5, 4)
 

In our daily life, we come across many situations, in which we can convert our problem into word problems and then we can take the help of algebra to solve them. Elementary algebra helps:
1.    To identify what components of an equation are constants and what are the changes in a situation, this represents the set of variables.
2.    To represent a pattern using a variable.
3.    To express a situation using words, equations, tables, and graphs etc.
4.    To identify and understand the relationship among the various forms of expression.
5.    To use objects from our daily life, to model the algebraic expressions.
6.    To translate an equation or algebraic expression into a situation, that it could represent.

Elementary algebra helps to solve an equation where we don’t know all of the information, or where one of the components of the equation varies or changes?
For example, by introducing a symbol to the unknown variable, lets say x, and then substituting the numbers for this letter, we can solve the expression x + 8,
as if x = 3, then the value of the expression would be 3 + 8 = 11.
If x + 8 = 6, then the value of the x would be 6 - 8 = -2 and so on.

Usually, we can connect the words with the number sentences, first without letters, and then with letters, which shows the elementary algebra is used in almost every part of our day to day life.