Lesson Types of systems - inconsistent, dependent, independent

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Source code of 'Types of systems - inconsistent, dependent, independent'

This Lesson (Types of systems - inconsistent, dependent, independent) was created by by mathick(4)

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About mathick:

This lesson concerns systems of two equations, such as:

The equations can be viewed algebraically or graphically. Usually, the problem is to find a solution for x and y that satisfies both equations simultaneously. Graphically, this represents a point where the lines cross. There are 3 possible outcomes to this (shown here in blue, green, and red):
<hr width = 80%>

The two lines might not cross at all, as in

<center>{{{graph( 300, 200, -20, 20, -20, 20, x, x+10) }}}

y = x
y = x + 10.</center>

This means there are no solutions, and the system is called inconsistent.

If you try to <a href = "http://www.mathick.com/mathick.php?topic=system">solve this system algebraically</a>, you'll end up with something that's not true, such as 0 = 10.

Whenever you end up with something that's not true, the system is inconsistent.

<hr width = 80%>
The two equations might actually be the same line, as in

<center>{{{graph( 300, 200, -20, 20, -20, 20, x+10, x+10) }}}
y = x + 10
2y = 2x + 20.</center>

These are equivalent equations. The lines are actually the same line, and they 'cross' at infinitely many points (every point on the line). In this case, there are infinitely many solutions and the system is called dependent.

If you try to <a href = "http://www.mathick.com/mathick.php?topic=system">solve this system algebraically</a>, you'll end up with something that's true, such as 0 = 0.

Whenever you end up with something that's true, the system is dependent.

<hr width = 80%>

The two lines might cross at a single point, as in

<center>{{{graph( 300, 200, -20, 20, -16, 24, x+10, 2x) }}}
y = x + 10
y = 2x.</center>

If you try to <a href = "http://www.mathick.com/mathick.php?topic=system">solve this system algebraically</a>, you'll end up with something that involves one of the variables, such as x = 10. In this case, there is just one solution, and the system is called independent.

Whenever you end up with something that involves one of the variables, such as x = 10, the system is independent.

<hr width = 80%>
Here are a couple of handy tables for recognizing what type of system you're dealing with.

You can try practice problems <a href = "http://www.mathick.com/mathick.php?topic=system">here</a>.

From the algebraic perspective:

<center><table border="1" cellspacing="0" cellpadding="0"> <tr> <td width="207" valign="top"> If solving using the addition or substitution method leads to </td> <td width="200" valign="top"> then the system is </td> <td width="184" valign="top"> and the equations </td> </tr> <tr> <td width="207"> X = a number, y = a number </td> <td width="200"> independent </td> <td width="184"> will have different values of m when both are placed in y = mx + b (slope-intercept) form </td> </tr> <tr> <td width="207"> an inconsistent equation, such as 0 = 3 </td> <td width="200"> inconsistent </td> <td width="184"> will have the same value of m, but different values of b, when both are placed in y = mx + b form </td> </tr> <tr> <td width="207"> An identity, such as 5 = 5 </td> <td width="200"> dependent </td> <td width="184"> will be identical when both are placed in slope-intercept form </td> </tr> </table></center>

From the graphical perspective:

<center><table border="1" cellspacing="0" cellpadding="0"> <tr> <td width="207" valign="top"> If the equations have </td> <td width="200" valign="top"> then the system is </td> <td width="184" valign="top"> and the lines </td> </tr> <tr> <td width="207"> Different slopes </td> <td width="200"> independent </td> <td width="184"> cross at a point </td> </tr> <tr> <td width="207"> the same slope but different intercepts </td> <td width="200"> inconsistent </td> <td width="184"> are parallel and never cross </td> </tr> <tr> <td width="207"> the same slope and the same intercept </td> <td width="200"> dependent </td> <td width="184"> are actually both the same line </td> </tr> </table></center>