Definition: Given 2 linear equations in 2 unknowns (variables). Finding a solution - that is, values for both variables that make both equations true - is called solving a 2х2 system of linear equations.

Example of a linear system of 2 equations in 2 unknowns.

Sample 2x2 linear system: \left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{4x - 2y = 0} \\ \end{array} \right.

The solution to this system is   (x,y)=(1,2) .
To see this we substitute \color{purple}{x=1} and \color{teal}{y=2} and check whether both equations are "true":
\begin{array}{c} \color{purple}{1}\color{blue}{+}\color{teal}{2}\color{blue}{ \cdot 2 \,≟\, 5} \\ \color{blue}{1 + 4 \,≟\, 5} \\ \color{blue}{5 \,=\, 5} \\ \end{array}      \begin{array}{c} \color{red}{4\cdot} \color{purple}{1} \color{red}{- 2\cdot} \color{teal}{2}\color{red}{ \,≟\, 0} \\ \color{red}{4 - 4 \,≟\, 0} \\ \color{red}{0 = 0} \\ \end{array}
Graphically, we have:

Regulation: A solution to the system is every intersection (touching) point of the 2 lines.

InterActivity   Directions for InterActivity

1. Look at the two lines a and b. They intersect at the point E. Check that E satisfies both equations.
If you know how, solve the system: a and b and check that you get E..

2. Click and drag the lines or the points A, B, C or D. If they intersect at one point, E is "good".

Look at the left (where the formulas are) to check that E is a point!

3. Click and drag the points A, B, C or D so that a and b are parallel. They are parallel when E says "undefined".

Notice that a and b are the same except for the constant after the "=" sign.

4. Click and drag the points A, B, C or D so that a and b coincide.

Notice that a and b are completely the same and that E is undefined.

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Solutions to 2x2 linear systems

A linear system  

  • The system has exactly one solution, i.e. the lines intersect in exactly one point and the solution is that point.
  • The system has no solution, i.e. the lines are parallel and never touch.
  • The system has infinitely many solutions, i.e. the lines coincide (are one and the same) and every point on this line is a solution.
Exactly one solution: \left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{4x - 2y = 0} \\ \end{array} \right. No solution: \left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{2x + 4y = 6} \\ \end{array} \right. Infinitely many solutions: \left\{ \begin{array}{c} \color{blue}{\,\,x + 2y = 5} \\ \color{red}{2x + 4y = 10} \\ \end{array} \right.

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Page last modified on September 27, 2008, at 04:42 AM