Equations Introduction

Definition

An equation is a mathematical statement denoting the equality of two expressions. Some examples of equations are $y=7x+2,\ x^2+y^2+xy=1,\ y=\sin(x)$ etc.

The LHS(Left Hand Side) and RHS(Right Hand Side) of equations like $y,\ x+2,\ x^2+y^2,\ 1$ etc are known as expressions. Expressions are combinations of variables, constants and operators.

The alphabetical values like $x,y$ are called variables and numerical values like $1,2$ etc are called constants.

The $+,\ -,\ \times,\ pow,\ \sin,\ \cos$ etc are called operators. Operators take operand as input. Eg, in the expression: $x+y$, $x$ and $y$ are operands and $+$ is operator. Operators that take two operands like $+,-,\times$ etc are called as binary operators. Similarly operators that take only one operand like $\sin, \cos$ etc are called as unary operators.

Classification of Equations

There are various criteria for classifying equations. Some of the criteria are explicitness, coordinate system, linearity/non-linearity etc. We discuss few of them as follow:

On Basis of Explicitness

Explicit Equation

An equation is said to be explicit if the output variable can be written as a function of the input variables. For example, if there is one variable $x$ then $y=f(x)$ is explicit. Similarly, if there are two input variables $x_1$ and $x_2$ then $y=f(x_1, x_2)$ is explicit. Examples are $y=x^2+x+1, x=y+8$ etc.

Implicit Equation

An equation is said to be implicit if the output variable can't be written as a function of input variables. These equations can be reduced to the the form $f(x,y)=0$ but not the the form $y=f(x)$. Examples are $x^2+y^2=1, xy+sin(x)=1$ etc.

On Basis of Coordinate System

Cartesian Equations

Equations written for the cartesian coordinate system are cartesian equations. Example $x^2+y^2=1, x+y=1, y=x, x^2+y^2=z$ etc.

Polar Equations

Equations written for the polar coordinate system are polar equations. Example $r=\theta, r^2+\theta^2=1$ etc.

Parametric Equations

Parametric equations define a separate function for each coordinate in terms of another parameter.

For 2D, there can be one parameter in terms of which $x$ and $y$ are defined. Example: $x=f_x(t), y=f_y(t)$ represents a curve in 2D. The shape of the curve is determined by functions $f_x$ and $f_y$ while the length of the curve is determined by the range of parameter $t$.

For 3D, there can be one or two parameters in terms of which $x,y,z$ are defined. Example: $x=f_x(u,v),\ y=f_y(u,v),\ z=f_z(u,v)$ represents a surface in 3D. In this case the shape of the surface is determined by $f_x$, $f_y$ and $f_z$ while the area is determined by the ranges of parameters $u,v$. When there is single parameter like $x=f_x(t),\ y=f_y(t),\ z=f_z(t)$, it also represents a curve in 3D.

On Basis of Field

Transcendental Equations

Equations that involve trigonometric, hyperbolic, logarithmic or exponential functions are called transcendental equations. Examples: $y=\sin(x), y=x\sin(x)$ etc.

Algebraic Equations

Equations involving only algebraic variables without trigonometric, hyperbolic, logarithmic or exponential functions are called algebraic equations. Examples: $x+y=1, x^2+y^2=2$ etc.

On Basis of Linearity

Linear Equations

The equation which represents a line in 2D or a plane in 3D or a hyper plane in higher dimension is called a linear equation. Examples of linear equations are $x+y+1=0, x+y+z+10=0$ etc.

Non-linear Equation

Any algebraic or transcendental equation that doesn't represent a line or a plane or a hyper plane is called a non-linear equation. Examples are: $x^2+y^2=1, y=x^2, y=\sin(x), y=x\tan(x)$ etc.