Lesson Types of systems - inconsistent, dependent, independent

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This lesson concerns systems of two equations, such as:

2x + y = 10
3x + y = 13.

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):

The two lines might not 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 = x
y = x + 10.

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

If you try to solve this system algebraically, 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.

The two equations might actually be the same line, as in

graph%28+300%2C+200%2C+-20%2C+20%2C+-20%2C+20%2C+x%2B10%2C+x%2B10%29+

y = x + 10
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, there are infinitely many solutions and the system is called dependent.

If you try to solve this system algebraically, 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.

The two lines might cross at a single 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 try to solve this system algebraically, 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.

Here are a couple of handy tables for recognizing what type of system you're dealing with.
You can try practice problems here.

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 have different values of m when both are placed in y = mx + b (slope-intercept) form
an inconsistent equation, such as 0 = 3	inconsistent	will have the same value of m, but different 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 slope-intercept form

From the graphical perspective:

If the equations have	then the system is	and the lines
Different slopes	independent	cross at a point
the same slope but different intercepts	inconsistent	are parallel and never cross
the same slope and the same intercept	dependent	are actually both the same line

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