If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Solutions to systems of equations: dependent vs. independent

A dependent system of equations has infinite solutions, and an independent system has a single solution. Watch an example of analyzing a system to see if it's dependent or independent. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• I understand that dependent systems have an infinite amoun of solutions and independent ones only have one solution, but why are they called that way? What is the logic behind these different classifications?
• According to Wikipedia (https://en.wikipedia.org/wiki/System_of_linear_equations#Independence), "independent" means that none of equations in a system of equations can be derived from each other.

Say you have these two equations:
``3x + 2y = 66x + 4y = 12``

then these equations would be dependent, since we can derive the second one from the first, or vice versa (by either multiplying the first one by a factor of two, or dividing the second one by a factor of two). Although no explanation is given for the choice of words "dependent" and "independent", maybe you can somehow relate these terms to this description.
• I may be jumping ahead a bit here, but what if we deal with curves that intersect twice?
• You are jumping ahead, but that's a good thing, not a bad thing.

These curves are like parabolas (a kind of U-shape on a graph), a type of conic section. When a line goes through a parabola in such a way that it intersects with the parabola twice, there are two different answers that the system has.
• Is the inconsistent graph independent or dependent?
• Within consistent graphs, there are dependent and independent equations. So, inconsistent graphs are neither dependent or independent.
• How do you when to use substitution or elimination?
• well, when you solve for "Y" with y=mx +b to determine whether or not your lines are running parallel to each other, and you determine that they ARE NOT and that there IS A SOLUTION or SOLUTIONS, then it is easy to SUBSTITUTE because you have already solved for "y" with y=mx+b.

You can use ELIMINATION when you ALREADY KNOW that your lines have a solution. For example, if you see a picture.

If you find y=mx+b and determine that the lines ARE running parallel to each other, then there is no need to try and solve the equations, THERE IS NO SOLUTION.

So, either one works. Knowing both Substitution and Elimination helps you not to try and solve your equations only one way. Thereby helping you work less.

• how can you determine the solution directly without using any graph ??
(its our lesson in math now)
• It all becomes clear if they are in the same form. It doesn't matter which, but let's get them into y = mx + c form.

Eq1) 4x + 2y = 16
Eq2) y = -2x + 8

Well, Eq2 is already in y = mx + c form. Let's convert Eq1:
4x + 2y = 16
2y = -4x + 16
y = -2x + 8

Aha! This looks suspicious! We have THE SAME EQUATION TWICE! So this system is dependant and consistent. But that's great for this example, can we make a general rule? Sure can!

If the coefficients (the 'm's) are different, that means the slopes are different. That means that the two lines are not parallel, and so they must meet eventually. It might be at a huge value of x or y, but they will meet! So:

Rule 1: If the slopes (the 'm's) are different, the system is independent (and therefore also consistent)

If the slopes are the same, the lines must either be on top of each other, or parallel. If they are on top of each other, the equations will be the same, so they will also have the same intercept (the 'c'). That means:

Rule 2: If the slopes (the 'm's) are the same, and the intercepts (the 'c's) are the same also, the system is dependent.

If the equations are parallel but not the same they must be paralle, but not on top of each other. Therefore:

Rule 3: If the slopes are the same, but the intercepts aren't (the 'c's), the system is inconsistent.

So, step 1: convert to y = mx + c form, step 2: apply the above three rules.

Hope that helps :)
• what happens in the graph if the two lines are not parallel but one is slightly tilted. Is that an inconsistent or do you just have to increase your graph to mark the intercept
• If the lines are not parallel, then they will eventually intersect; therefore, it will have a solution.
• If the graph of a system of linear equations shows three lines such that each pair of lines intersects at a point different from the intersection point for any other pair, how many solutions does the system have?
• It means there are no solutions for the system. A solution would be a point where all three lines intersect. The points of intersection between two lines are valid solutions for that part of the system (you could substitute it into these equations and it would be valid), but not the overall system.
• Why did Sal not substitute in the y equation?
• Part of it was based on what the question was asking - is it consistent or inconsistent. If you would have substituted and took it to completion, you would end up with 0 = 0.
• At around , if Consistent solution #1 is independent, Consistent solution #2 is dependent, then what is an INconistent solution?
• He draws an inconsistent system at - it's 2 parallel lines.