The paper analyzes and discusses different concepts related to Algebra. The paper talks
about Order of Operations, Linear equations and their application, linear inequalities,
Quadratic equations. It talks about and summarizes different methods of tackling and
simplifying such algebraic expressions. Finally the paper dwells into different functions and
their graphic representations.
Algebra and Functions
Order of Operations: Order of operations is a mathematical that determines the
correct order for solving any of math operations. Powers and roots are solved before
multiplication and division, which in turn are solved before addition and subtraction
(Toolingu, n.d.). The rule is used in calculation and simplification of a complex algebraic
expression involving many arithmetic operations. Order of operations is formed of set of
three rules:
Rule 1: All calculations and simplifications inside parenthesis are to be done first.
Rule 2: Then all the multiplications and divisions are to be done from left to right and
division has to be given precedence over multiplication.
Rule 3: In the end all the adds and subtracts are to be done.
Linear Equation in one variable: Linear equation in one variable is described as an
equation with only one variable of highest power one or of first order. Such equations only
have one solution or root. E.g.
3x+5=x+7 is a linear equation in one variable
Linear equations in one variable have huge number of applications and are used in
calculation of various types of problems. Some of the most common and prevalent
applications of Linear equations in one variable are:
1) They are used in solving problems involving relationships between real numbers.
2) They are used in geometry for problems involving calculation of perimeter or sides.
3) They are very commonly used in problems involving Money and Percents.
4) They are used extensively in time, distance and speed problems.
Formula/ General method to solve Linear Equations:
1) Clear Fractions: It involves multiplying both sides of the equation by Least Common Denominator.
2) Simplifying each side separately: This step uses distributive property to clear parentheses and combine like terms.
3) Isolating variable terms on one side
4) Substituting the proposed solution into the original equation.
This formula is a generic and can be used to solve all the linear equations in one variable.
Linear Inequalities: A linear inequality involves a linear expression in two variables by
using any of the relational symbols such as <, >, <= or >= (IcoachMath, n.d.). A linear
equality divides a plane in two parts based on the solutions of the equation. Some of the
examples of linear inequalities are:
x+6>=14 2x-3y<14 3a +7b<=78 a +2b>76
Linear Inequalities in Two variables: Linear equalities which involve two variables or
two unknowns. The solution of a linear equality in two variables like Ax +By> C is an
ordered pair (x, y) that produces a true statement when the values of x and y are
substituted into the inequality (MathPlanet, n.d.). E.g.
2x + 5y>6 is a Linear inequality with x and y as two variables.
Solving Quadratic Equations using factoring: Quadratic equation is an equation in which
the highest power of an unknown quantity is a square (Princeton, n.d.). A general quadratic
equation can be written as ax^2+bx+c=0.
Solving quadratic equations using factoring involves factoring of the coefficient of x term
(b) in the equation in such a manner that the multiplication of those two factors yield the
third coefficient (c). Before the method is applied it has to be ensured that the coefficient of
x^2 term is 1. The process can be explained with a simple example:
Quadratic Equation: x^2+5x+6=0, where a=1, b=5 and c=6.
The equation can be written as: x^2+ 2x+3x+6=0; 5 has been factored to ensure that the multiplication of terms gives 6.
Final step: x (x+2) +3(x+2); x=-2 and x=-3 are the two solutions.
Solving Quadratic Equations using Formula: This is the second and more commonly
used method to find solutions for quadratic equations. This method uses a formula to
calculate both roots of the equations simply by substituting values of a, b and c from the
general quadratic equation i.e. ax^2+bx+c=0. According to the formula the value of roots
for general quadratic equation is given by:
Functions and Graphs: A function is a relation, most probably an equation, in which no
two ordered pairs have same x-coordinate when graphed. Functions are evaluated using
graphs in most of the cases and hence both are studied together. There are various types of
functions like even functions, odd functions, mod functions etc. which when plotted on a
graph give us standard shapes while there are certain other functions where the graph
depends on the values of variables. One can infer about the kind of equation a function is
graphs is as shown below:
Figure 1. Functions. Reprinted from Straight Line Graphs,
In Tutornext, n.d., Retrieved November 12, 2012,
raight-line-graphs.html.
ICoachMath (n.d.), Linear Inequality. Retrieved 12 November, 2012, from
http://www.icoachmath.com/math_dictionary/Linear_Inequality.html
MathPlanet (n.d.), Linear Inequalities in two variables. Retrieved 12 November, 2012, from
http://www.mathplanet.com/education/algebra-1/linear-inequalitites/linear-inequalities-in-two-variables
Princeton (n.d). Retrieved 12 November, 2012, from
http://wordnetweb.princeton.edu/perl/webwn
Toolingu (n.d.), Math Fundamentals: 100. Retrieved 12 November, 2012, from
http://www.toolingu.com/definition-800100-11627-order-of-operations.html
