Remember how the solutions to a system of equations are found graphically, and analyze the structure of the equations forming the system. What can you do to have an inconsistent system?
Example Solution:
x+y+z =7 y+z =4 y+z =8
Practice makes perfect
Let's start by reviewing how we can solve a system of equations by graphing. When graphing a system of three equations with three variables, each equation represents a plane in a coordinate space. The intersections of these planes are the solutions to our system. Now, let's have a look at our system.
x+y+z =7 & (I) y+z = & (II) y+z = & (III)
For this system to have no solutions, we need to make sure that there are no points where the three of planes intersect. Since the left-hand side of Equations (II) and (III) are the same, we can choose the right-hand side of them to be two different constants.
x+y+z=7 & (I) y+z = 4 & (II) y+z = 8 & (III)
With this choice Equations (II) and (III) are inconsistent. Notice that they have the structure of two linear equations representing parallel lines in two dimensions. In a coordinate space, they represent two parallel planes.
Even if the parallel planes intersect the other one, there is no point at which all three planes intersect. Therefore, the system has no solution.
Notice that this is not the only way of having an inconsistent system of three equations. There are infinitely many possible solutions satisfying this requirement, and this is only an example solution.
