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  • MAT223 Lecture 02

Consistent and Inconsistent Linear Equations

• Consistent equations have at least one solution.
• Inconsistent equations have no solutions.
• Consistent
  • $$\begin{array}{c}x+y=3\\ x-y=1\end{array}$$
• Inconsistent
  • $$\begin{array}{c}x+y=4\\ x+y=1\end{array}$$
  • No solutions
• For MAT223 you cannot simply define a domain for your solutions and move the solutions outside the domain to make a consistent system become inconsistent.
• Ex:
  • $y=2-x$
  • $y=x-2$
  • Not inconsistent because these lines cross.
• Ex:
  • Turn a consistent system into an inconsistent system by changing one coefficient.
  • $$\begin{array}{c}x+y=3\\ x-y=1\end{array}$$
  • $$\begin{array}{c}x+y=3\\ x+y=1\end{array}$$
• Ex:
  • Turn a consistent system into an inconsistent system by changing exactly one constant term.
  • $$\begin{array}{c}x+y=3\\ x-y=1\end{array}$$
  • $y=-x+3$
  • $y=x-1$
  • Not possible to do it, changing y-intercept won't let us change the fact these lines intersect.