Consistent and Inconsistent Systems
Identify consistent and inconsistent systems of equations in Grade 10 algebra. Determine if a system has at least one solution (consistent) or no solution (inconsistent) from graphs and algebra.
Key Concepts
A system that has at least one solution is consistent . A system of simultaneous equations that have no common solution is inconsistent .
If elimination leads to a true statement like $0=0$, the system is consistent with infinite solutions. If elimination leads to a false statement like $0=10$, the system is inconsistent because parallel planes never intersect, such as in the equations $x+y+z=5$ and $2x+2y+2z=15$.
Think of it like planning with friends! A 'consistent' system is when everyone agrees on a meeting spot (one point) or a path to walk together (a line). An 'inconsistent' system is when your friends' plans are impossible to sync up—they're on different parallel paths and will never, ever meet. No solution means no fun!
Common Questions
What is the difference between consistent and inconsistent systems?
A consistent system has at least one solution (lines intersect or coincide). An inconsistent system has no solution (parallel lines that never meet).
How do you identify an inconsistent system algebraically?
When solving, if variables cancel and produce a false statement (e.g., 3 = 7), the system is inconsistent. The equations have no common solution.
How do parallel lines on a graph indicate an inconsistent system?
Parallel lines have equal slopes but different y-intercepts. They never intersect, so there is no point satisfying both equations simultaneously — the system is inconsistent with no solution.