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| − | + | Linearity! A concept that is easy to understand with the right definition! | |
| − | + | A system may be defined as linear if for every independent variable value 'x' put into the system, representing by equation y = f(x), there is a unique dependent variable value 'y.' | |
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| − | + | An example of a linear system is a straight line with slope m = 1 represented by the equation y = x. An example of a non-linear system would be a system represented by the equation y = x^2. | |
| − | + | An easy way to prove if a system is linear is to plot the equation representing the system and then draw a horizontal line, i.e. y = 1, and observe at how many points the line crosses through the equation. If the two lines intersect only once for all values of x then the system is linear. Otherwise, if the two equations intersect at more than one location then the system is non-linear. Indeed, the equations y = x and y = 1 will intersect only once at (x,y) = (1,1), while the equations y = x^2 and y = 1 will intersect at the locations (-1,1) and (1,1). These results prove that y = x represents a linear system and y = x^2 represents a non-linear system! | |
Latest revision as of 13:16, 12 September 2008
Linearity! A concept that is easy to understand with the right definition!
A system may be defined as linear if for every independent variable value 'x' put into the system, representing by equation y = f(x), there is a unique dependent variable value 'y.'
An example of a linear system is a straight line with slope m = 1 represented by the equation y = x. An example of a non-linear system would be a system represented by the equation y = x^2.
An easy way to prove if a system is linear is to plot the equation representing the system and then draw a horizontal line, i.e. y = 1, and observe at how many points the line crosses through the equation. If the two lines intersect only once for all values of x then the system is linear. Otherwise, if the two equations intersect at more than one location then the system is non-linear. Indeed, the equations y = x and y = 1 will intersect only once at (x,y) = (1,1), while the equations y = x^2 and y = 1 will intersect at the locations (-1,1) and (1,1). These results prove that y = x represents a linear system and y = x^2 represents a non-linear system!
