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.

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
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\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.

Global	2x2 System of Equations
Brief	Mathcasts and Interactivities to understand and practice solving systems of 2 equations in 2 variables (unknowns).
Grade	7-10 Interactivities start at 8th grade level on up
Strand	Algebra; Expressions, Equations and Inequalities
Standard	Algebra 1 3.1, Algebra 1 3.2, Algebra 1 3.3, ACT EE 28-32
Keywords	system, linear system, variable, unknown, linear equation, elimination, addition, substitution
Comments	none
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Author	LFS - contact - website
Type	Freeware - Available for Offline and Online Use - Translatable (html)
Use	Requires sunJava player