How to determine whether a system of equations is independent, dependent, or inconsistent

You can determine whether a system of equations is independent, dependent, or inconsistent in two ways, either graphically or algebraically.


A) Graphically

• When the equations' graphs intersect at a single point, as follows, they are called independent.
• When the equations' graphs coincide at every point - that is, both equations have the same graph, they are called dependent.
• When the equations' graphs do not intersect at any point - that is, they are parallel lines, the system is said to be inconsistent.

B) Algebraically

• When on writing each equation in its slope intercept form, you get the same equations, that is, when all the given equations are essentially the same equation, the system is known as dependent. On solving such a system of equation, you will get solutions like 0 = 0, 1 = 1, which are known as identities. This implies that the equations are true for all values of all variables in them, because they are the same equations.
• When, on solving the system of equations, you get solutions for each variable, the system is independent.
• When, on solving the equation, you get no solution, that is, something like 0 = 1, 10 = 20, then the system is said to be inconsistent.