Linear Systems in 2 Variables

a) Consistent Independent System 

3x-4y=-6                     
2x+4y=16                                                                             
  

*This is a system with only one solution. 
*The graphical solution is the intersection of two straight lines. 







b) Consistent Dependent System 

x+y=1                 -2x+2y=-2     *This is a system with infinite 
2x+2y=2               2x+2y=2        solutions. 
              
                      

*The graphical solution is any point on the two identical straight lines or coincident lines.                       







This is again a Consistent Dependent System. 
2x+2y=6 
m= -1 
b= 3 
x+y=3          

c) Inconsistent System 
     x+y=3    *This is a system with no solution. 
     x+y=1

*The graphical solution has parallel lines. 





*If the equations have different slopes then the system is independent and the lines cross at a point. 
*If the equations have the same slope but different intercepts, then the system is inconsistent and the lines are parallel and never cross. 
*If the equations have the same slope and the same intercept, then the system is dependent and the lines are actually both in the same line.