Inconsistent Consistent Dependent Consistent Independent

This Lesson (Types of systems - inconsistent, dependent, independent) was created by by mathick(4) About Me : View Source, Show
About mathick:

This lesson concerns systems of 2 equations, such as:

2x + y = 10
3x + y = xiii.

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 cantankerous. There are three possible outcomes to this (shown here in blue, green, and reddish):

The two lines might non cross at all, as in graph%28+300%2C+200%2C+-20%2C+20%2C+-20%2C+20%2C+x%2C+x%2B10%29+

y = ten
y = 10 + 10.

This ways at that place are no solutions, and the system is chosen inconsistent.

If you attempt to solve this system algebraically, you'll finish upwards with something that'southward not true, such as 0 = 10.

Whenever you lot end up with something that's not truthful, the organisation is inconsistent.

The two equations might really be the same line, equally in graph%28+300%2C+200%2C+-20%2C+20%2C+-20%2C+20%2C+x%2B10%2C+x%2B10%29+

y = x + x
2y = 2x + 20.

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, at that place are infinitely many solutions and the system is called dependent.

If you try to solve this arrangement algebraically, you'll end upwardly with something that's true, such as 0 = 0.

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

The ii lines might cantankerous at a unmarried point, as in graph%28+300%2C+200%2C+-20%2C+20%2C+-16%2C+24%2C+x%2B10%2C+2x%29+

y = x + 10
y = 2x.

If you endeavour to solve this system algebraically, you'll end upwardly with something that involves one of the variables, such every bit ten = 10. In this case, in that location is just i solution, and the organisation is called independent.

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

Here are a couple of handy tables for recognizing what type of arrangement you're dealing with.
You can try practice bug hither.

From the algebraic perspective:

If solving using the addition or substitution method leads to	then the system is	and the equations
X = a number, y = a number	independent	will take unlike values of chiliad when both are placed in y = mx + b (slope-intercept) form
an inconsistent equation, such equally 0 = three	inconsistent	will accept the same value of m, but dissimilar values of b, when both are placed in y = mx + b form
An identity, such as 5 = 5	dependent	will be identical when both are placed in gradient-intercept form

From the graphical perspective:

If the equations accept	and then the organisation is	and the lines
Unlike slopes	independent	cross at a point
the aforementioned slope merely different intercepts	inconsistent	are parallel and never cross
the same slope and the same intercept	dependent	are actually both the same line