Main content

## Algebra 1

### Course: Algebra 1 > Unit 6

Lesson 5: Number of solutions to systems of equations- Systems of equations number of solutions: fruit prices (1 of 2)
- Systems of equations number of solutions: fruit prices (2 of 2)
- Solutions to systems of equations: consistent vs. inconsistent
- Solutions to systems of equations: dependent vs. independent
- Number of solutions to a system of equations
- Number of solutions to a system of equations graphically
- Number of solutions to a system of equations graphically
- Number of solutions to a system of equations algebraically
- Number of solutions to a system of equations algebraically
- How many solutions does a system of linear equations have if there are at least two?
- Number of solutions to system of equations review

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# 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?(24 votes)
- 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 = 6`

6x + 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.(4 votes)

- I may be jumping ahead a bit here, but what if we deal with curves that intersect twice?(10 votes)
- 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.(16 votes)

- Is the inconsistent graph independent or dependent?(5 votes)
- Within consistent graphs, there are dependent and independent equations. So, inconsistent graphs are neither dependent or independent.(12 votes)

- How do you when to use substitution or elimination?(6 votes)
- 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.

I hope this was helpful!(5 votes)

- how can you determine the solution directly without using any graph ??

(its our lesson in math now)(4 votes)- 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 :)(8 votes)

- 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(3 votes)
- If the lines are not parallel, then they will eventually intersect; therefore, it will have a solution.(6 votes)

- 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?(3 votes)
- 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.(4 votes)

- Why did Sal not substitute in the y equation?(3 votes)
- 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.(4 votes)

- At around0:06, if Consistent solution #1 is independent, Consistent solution #2 is dependent, then what is an INconistent solution?(3 votes)
- He draws an inconsistent system at1:20- it's 2 parallel lines.(3 votes)

- would the parallel lines also be independent?(2 votes)
- No, the parallel lines are inconsistent.

Inconsistent lines are not independent or dependent, just inconsistent.

But consistent lines can be either dependent or independent.(5 votes)

## Video transcript

Is the system of
linear equations below dependent or independent? And they give us two
equations right here. And before I tackle
this specific problem, let's just do a little
bit a review of what dependent or independent means. And actually, I'll compare that
to consistent and inconsistent. So just to start
off with, if we're dealing with systems of linear
equations in two dimensions, there's only three possibilities
that the lines or the equations can have relative to each other. So let me draw the
three possibilities. So let me draw three
coordinate axes. So that's my first
x-axis and y-axis. Let me draw another one. That is x and that is y. Let me draw one
more, because there's only three possibilities
in two dimensions. x and y if we're dealing
with linear equations. So you can have the situation
where the lines just intersect in one point. Let me do this. So you could have
one line like that and maybe the other line
does something like that and they intersect at one point. You could have the
situation where the two lines are parallel. So you could have a situation--
actually let me draw it over here-- where you have one
line that goes like that and the other line has the
same slope but it's shifted. It has a different y-intercept,
so maybe it looks like this. And you have no points
of intersection. And then you could
have the situation where they're actually the same
line, so that both lines have the same slope and
the same y-intercept. So really they
are the same line. They intersect on an
infinite number of points. Every point on
either of those lines is also a point
on the other line. So just to give you a little
bit of the terminology here, and we learned this in the
last video, this type of system where they don't intersect,
where you have no solutions, this is an inconsistent system. And by definition, or
I guess just taking the opposite of
inconsistent, both of these would be considered consistent. But then within consistent,
there's obviously a difference. Here we only have one solution. These are two different lines
that intersect in one place. And here they're essentially
the same exact line. And so we differentiate
between these two scenarios by calling this one
over here independent and this one over
here dependent. So independent-- both lines
are doing their own thing. They're not dependent
on each other. They're not the same line. They will intersect
at one place. Dependent-- they're
the exact same line. Any point that satisfies one
line will satisfy the other. Any points that
satisfies one equation will satisfy the other. So with that said, let's see if
this system of linear equations right here is dependent
or independent. So they're kind of
having us assume that it's going
to be consistent, that we're going to
intersect in one place or going to intersect in an
infinite number of places. And the easiest way to
do this-- we already have this second equation here. It's already in
slope-intercept form. We know the slope is negative
2, the y-intercept is 8. Let's put this first equation
up here in slope-intercept form and see if it has a different
slope or a different intercept. Or maybe it's the same line. So we have 4x plus
2y is equal to 16. We can subtract 4x
from both sides. What we want to do is isolate
the y on the left hand side. So let's subtract
4x from both sides. The left hand side-- we
are just left with a 2y. And then the right hand side,
we have a negative 4x plus 16. I just wrote the negative
4 in front of the 16, just so that we have it in the
traditional slope-intercept form. And now we can divide both
sides of this equation by 2, so that we can isolate
the y on the left hand side. Divide both sides by 2. We are left with y is equal
to negative 4 divided by 2 is negative 2x plus
16 over 2 plus 8. So all I did is algebraically
manipulate this top equation up here. And when I did that, when
I solved essentially for y, I got this right
over here, which is the exact same thing
as the second equation. We have the exact same slope,
negative 2, negative 2, and we have the exact
same y-intercept, 8 and 8. If I were to graph these
equations-- that's my x-axis, and that is my y-axis-- both
of them have a y-intercept at 8 and then have a
slope of negative 2. So they look
something-- I'm just drawing an approximation of it--
but they would look something like that. So maybe this is the graph
of this equation right here, this first equation. And then the second equation
will be the exact same graph. It has the exact
same y-intercept and the exact same slope. So clearly these two
lines are dependent. They have an infinite
number of points that are common to both of them,
because they're the same line.