Systems of Linear Equations - Inconsistent

 Definition: A system of linear equations is inconsistent if there is no solution.

Regulation: A 2x2 system is inconsistent if and only if the lines are parallel   See!

 No solution: \left\{ \begin{array}{c} \color{blue}{x + 2y = 5} \\ \color{red}{x + 2y = 2} \\ \end{array} \right. No solution: \left\{ \begin{array}{c} \color{blue}{y=3x+1} \\ \color{red}{y = 3x-3} \\ \end{array} \right. This browser does not have a Java Plug-in. Get the latest Java Plug-in here. This browser does not have a Java Plug-in. Get the latest Java Plug-in here.
Regulation: A system is inconsistent if -when solving- you get a stupid statement like 2=1    See & Hear!
The substitution method. \left\{ \begin{array}{c} \color{blue}{x + 2y = 5} \\ \color{red}{x+2y=2} \\ \end{array} \right.
 \left\{ \begin{array}{l} \color{blue}{x + 2y = 5} \\ \color{red}{x+2y=2} \\ \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} \color{blue}{x = 5-2y} \\ \color{navy}{x+2y=2} \\ \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} \color{navy}{x = 5-2y} \\ \color{red}{\color{blue}{(5-2y)} +2y =2} \\ \end{array} \right. \Leftrightarrow ---- \left\{ \begin{array}{l} \color{navy}{x = 5-2y} \\ \color{red}{5 = 2} \\ \end{array} \right. Second equation is stupid (untrue) \Leftrightarrow System is inconsistent.
The addition method. \left\{ \begin{array}{c} \color{blue}{x-y=1} \\ \color{red}{y - x =1} \\ \end{array} \right.
 \left\{ \begin{array}{l} \color{blue}{x-y=1} \\ \color{red}{y-x = 1} \\ \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} \color{navy}{x-y = 1} \\ \color{red}{-x + y = 1} \\ \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} \color{navy}{x -y=1} \\ \color{purple}{0+0=1} \\ \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} \color{navy}{x -y=1} \\ \color{purple}{0=1} \\ \end{array} \right. Second equation is stupid (untrue) \Leftrightarrow System is inconsistent.
 InterActivity   Directions for InterActivity 1. Look at the two lines a and b. They are parallel. They do not intersect. This system is inconsistent. Look in the left window. Notice that a and b are the same except for the constant after the "=" sign. So "simultaneously" we have "something=3" and the "same something=1". This is stupid. Mathematicians call stupid "inconsistent". (Remember - system means "simultaneous" solution - but here no point works in both lines at the same time.) The point E is the solution. When they are parallel when E says "undefined". 2. Click and drag the lines or the points A, B, C or D so that they intersect at a point E. Look in the left window. Check that E is now defined (a point). This system is consistent. It has a solution it has a solution.   3.Click and drag the points A, B, C or D so that a and b coincide. This system is consistent. It has infinitely many solutions. Look in the left window. Notice that a and b are completely the same. Here we have just repeated ourselves.  and that E is undefined. This browser does not have a Java Plug-in. Get the latest Java Plug-in here.

Related topics: