Inconsistent System
Recognize an inconsistent system of equations: when lines are parallel or algebra produces a false statement, the system has no solution and the equations are incompatible.
Key Concepts
Property A linear system with no solutions is an inconsistent system. The graphs of the equations are parallel lines that never intersect. When solving, you will arrive at a false statement, such as $0 = 64$.
Solve $\begin{cases} 3x 4y = 24 \\ 6x 8y = 16 \end{cases}$. Multiply the first equation by 2: $ 2(3x 4y) = 2( 24)$ gives $ 6x + 8y = 48$. Add the new equation to the second original equation: $( 6x + 8y) + (6x 8y) = 48 + 16$, which simplifies to $0 = 64$. Since $0=64$ is a false statement, the system has no solution and is inconsistent.
This is like getting a math puzzle that leads to a contradiction. If you do everything right but end up with a nonsensical statement like $0 = 10$, the system is telling you something important. The two lines are parallel and will never, ever cross. There is no point in the universe that can make both equations true.
Common Questions
What is an inconsistent system of equations?
An inconsistent system is a set of linear equations that has no solution. Graphically the lines are parallel and never intersect. Algebraically, solving produces a false statement such as 0=5, signaling no solution exists.
How do you identify an inconsistent system algebraically?
When you use substitution or elimination, all variables cancel and you are left with a false numeric statement like 3=7. This contradiction means the equations describe lines with the same slope but different y-intercepts.
What is the difference between an inconsistent system and a dependent system?
An inconsistent system has no solution (parallel lines). A dependent system has infinitely many solutions (same line). Elimination on a dependent system produces a true statement like 0=0, whereas inconsistent systems give a false statement.