The first pair is "consistent" versus "inconsistent."

Now, keep in mind that you are applying these to a system of linear

equations. We say that a point is a "solution" to the system when it

makes BOTH equations true, right? This is to say that there exists a

point (or set of points) that "work" in one equation and also "work"

in the other one. So we say that this point is CONSISTENT from one

equation to the next.

On the other hand, if there are NO points that work in both, then we

say that the equations are INCONSISTENT. NO numbers that work in one

are consistent with the other.

To sum up, a consistent system has at least one solution. An

inconsistent system has NO solution at all.

Now for the other pair. "Dependent" versus "Independent."

When a system is "dependent," it means that ALL points that work in

one of them ALSO work in the other one. Graphically, this means that

one line is lying entirely on top of the other one, so that if you

graphed both, you would really see only one line on the graph, since

they are imposed on top of each other. One of them totally DEPENDS

on the other one.

When a system is "independent," it means that they are not lying on

top of each other. There is EXACTLY ONE solution, and it is the point

of intersection of the two lines. It's as if that one point is

"independent" of the others.

To sum up, a dependent system has INFINITELY MANY solutions.

An independent system has EXACTLY ONE solution.

Now we can combine the terms. Since we have 2 terms and 2 terms, you

would think that there would be 4 possibilities:

1. Consistent Dependent

2. Consistent Independent

3. Inconsistent Dependent

4. Inconsistent Independent

However, 3 and 4 are not possible, since "Inconsistent" means no

solution. Independent and Dependent BOTH mean there is a solution, so

they can't ever go with Inconsistent because that would be

contradictory.

So really there are only three possibilities: Consistent Dependent,

Consistent Independent, and Inconsistent.

We ordinarily don't even use "consistent" with dependent or

independent, since once you know what these latter two words mean, you

already know they are consistent, so it is enough to say the system

is "dependent" or "independent."

We usually use the word "consistent" when we are more interested in

indicating that the system does HAVE a solution, rather than

indicating how many solutions it has.

you've got the definitions, but do you know what they mean?  

An inconsistent (or overdetermined) system is one without a solution.  

A dependent (or underdetermined) system is one where there is more 

than one solution.  Let's give some simple examples:

Inconsistent/overdetermined:

     (I1)  0 = 1

     (I2)  y = 1  and  y = 2

     (I3)  x = y  and  x = y + 1.

Dependent/underdetermined:

     (D1)  x = y

     (D2)  y = 3x  and  2y = 6x

     (D3)  x + 2y = 0.

you've got the definitions, but do you know what they mean?

An inconsistent (or overdetermined) system is one without a solution.

A dependent (or underdetermined) system is one where there is more

than one solution. Let's give some simple examples:

Inconsistent/overdetermined:

(I1) 0 = 1

(I2) y = 1 and y = 2

(I3) x = y and x = y + 1.

Dependent/underdetermined:

(D1) x = y

(D2) y = 3x and 2y = 6x

(D3) x + 2y = 0.

Now, some comments on each of these. (I1) is a false statement, and

even though it is not strictly a system, or even a statement about

unknown quantities (like x or y), it is inconsistent because it is

false! In (I2), it is a system, but it cannot be true, because it

implies 1 = 2. In (I3), we again have a similar situation; x = y = y+1

implies 0 = 1.

In (D1), there are infinitely many solutions, like x = y = 5, or

x = y = -1. In (D2), the second equation is the same as the first when

divided by 2, so it provides no additional information about x or y.

Finally, (D3) is the same as saying x = -2y, so it too is dependent

for the same reasons (D1) is.

Now, say you are given y = 3x + 5. You want to write another equation

which, with the given equation, makes an inconsistent system. There

are lots of ways to do this; one way is to say

y = 3x + 6,

so then we are forced to conclude that 3x + 5 = 3x + 6, or 5 = 6.

Even easier,

y = 3x

is also another possible way to do this. Now, suppose you want to make

a dependent system. Well, you can do it by following example (D2), and

multiply both sides by some number. 2 sounds good:

2y = 6x + 10.

Wow, that was pretty easy. Hmm... You could also do it another way, by

adding or subtracting some number from both sides:

2y - 4 = 6x + 6.

Now, some comments on each of these. (I1) is a false statement, and 

even though it is not strictly a system, or even a statement about 

unknown quantities (like x or y), it is inconsistent because it is 

false! In (I2), it is a system, but it cannot be true, because it 

implies 1 = 2. In (I3), we again have a similar situation; x = y = y+1 

implies 0 = 1. 

In (D1), there are infinitely many solutions, like x = y = 5, or 

x = y = -1. In (D2), the second equation is the same as the first when 

divided by 2, so it provides no additional information about x or y.  

Finally, (D3) is the same as saying x = -2y, so it too is dependent 

for the same reasons (D1) is.

Now, say you are given y = 3x + 5.  You want to write another equation 

which, with the given equation, makes an inconsistent system. There 

are lots of ways to do this; one way is to say

     y = 3x + 6,

so then we are forced to conclude that 3x + 5 = 3x + 6, or 5 = 6.  

Even easier,

     y = 3x

is also another possible way to do this. Now, suppose you want to make 

a dependent system. Well, you can do it by following example (D2), and 

multiply both sides by some number.  2 sounds good:

     2y = 6x + 10.

Wow, that was pretty easy. Hmm... You could also do it another way, by 

adding or subtracting some number from both sides:

     2y - 4 = 6x + 6.