Reflect on the concepts of linear and non-linear systems. What concepts (only the names) did you need to accommodate the concept of linear and non-linear systems in your mind? What are the simplest linear system and non-linear system you can imagine? In your day to day, is there any occurring fact that can be interpreted as linear systems and non-linear systems? What strategy are you using to get the graph of linear systems and non-linear systems?

Linear means something related to a line. All the linear equations are used to construct a line. A non-linear equation is such which does not form a straight line. It looks like a curve in a graph and has a variable slope value.

Linear Equations

•  A simple linear equation is of the form: y = mx + c
•  A linear equation looks like a straight line when graphed.
•  It has a constant slope value.
•  The degree of a linear equation is always 1.
•  Superposition principle is applicable to a system characterized by a linear equation.
•  The output of a linear system is directly proportional to its input.

Non-Linear Equations

•  A simple non-linear equation is of the form: ax2 + by2 = c
•  A non-linear equation look like a curve when graphed.
•  It has a variable slope value.
•  The degree of a non-linear equation is at least 2 or other higher integer values. With the increase in the degree of the equation, the curvature of the graph increases.
•  Superposition principle does not apply to the systems characterized by non-linear equations.
•  The input and output of a non-linear system is not directly related.

In mathematics, a linear function is one which satisfies both of the following properties:

• Additivity or superposition principle:  ?f(x+y)=f(x)+f(y);?

• Homogeneity:  ?f(αx)=αf(x).?

One of the simplest linear equation is y=x+2

An example for non linear equation is y=x² or y=x² +3x+ 1

Real life examples for linear system

You take a car for rent. They have a fixed charge of 150 plus 50 for every hour. This can be framed in a equation as y=50∗t+150 where t is hours used.

You are driving a car at the speed of 60km/hr. Distance covered by you after t hours of driving is y=60∗t

Real life examples for linear system

The distance travelled by a vertically thrown object in terms of time gives a non linear system.

Compound intrest also forms a non linear equation

To plot the graph the most common method is to substitute one value for x and find the corresponding y value. Create a table and mark the points. Then by joining the points we can plot the graph. Software and online tools can be used to plot the graphs of any equation.

To obtain the graphs of linear equations we just need 2 points. only a single line can pass through 2 points. Obtaining graph of a linear equation is simple while the for nonlinear system it is complicated.