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### Course: Class 10 (Foundation) > Unit 4

Lesson 6: (Bonus) Pair of equations in 2 variables# How many solutions does a system of linear equations have if there are at least two?

Sal answers this question for you! Created by Sal Khan.

## Want to join the conversation?

- I found a problem in the skill “Understanding systems of equations word problems” that I believe has an incorrect solution according to khan academy.

It asked the following: “You are solving a system of two linear equations in two variables, and you discover that there are no solutions to the system. Which of the following graphs could describe the system of equations?”

It asked me to select all the answers that apply. I selected only the blue graph and submitted my answer. It was incorrect. I then added the green graph to my answer and submitted. It was correct despite the fact that the green graph had only two vertical lines whose equations were x= -7 and x= -4. This is inconsistent with the statement that I was solving a system of "two linear equations in ***two*** variables” because the lines on the green graph only used the x variable.

Is the supposed solution actually erroneous or is there something about the problem I am failing to understand?(7 votes)- That's tricky. I see why you would feel cheated, because of the way the subject has been taught. The important thing to remember is that ANY two parallel lines will have no intersection point. And ANY line (vertical or otherwise) can be represented by a linear equation. And ANY linear equation IS an equation in two variables (whether both variables are explicitly written or not). For example, the equations you mention would be written as follows in standard form: x + 0y = -7, and x + 0y = -4

Because the coefficients on the y terms are zero, that term is not written. But it's still there.

Also what Arlic stated below about slope is not true, vertical lines have an undefined slope. Because there is no change in x for any change in y, the slope is: "change in y" divided by zero. Division by zero is undefined.(26 votes)

- What is the purpose of the video?(0 votes)
- I think it's to give us an example of linear equations(14 votes)

- Are there any more solutions beside "no solutions","infident solutions",and "one solution"? Just wondering!! (sorry for spelling errors)(5 votes)
- For linear equations, no, that's it. Because through any two points is a line. And then they'll have all the same points so infinite solutions. You can get more solutions with different types of functions but that's it for linear functions.(5 votes)

- In using the elimination method, say it was clear that one factor could be easily eliminated. Could I also set both equations equal to zero, then set them equal to one another, and eliminate this factor that way?(3 votes)
- Do you mean a set of equations like the following:

3x -2y = 5

5x-2y = 7

For this, yes, you could rewrite as:

3x - 2y - 5 = 0

5x - 2y - 7 = 0

and then set them equal:

3x - 2y - 5 = 5x - 2y - 7

Adding 2y to each side gives

3x - 5 = 5x - 7

Then we can solve for x by consolidating the x and constant terms:

-2x = -2

x = 1

Then we can use one of the original equations to solve for y

3 - 2y = 5

-2y = 2

y = -1

I hope this is what you meant...(8 votes)

- What is the difference between "Zero Solutions" and "Infinitely" many solutions? Like I know the answer is one of those two whenever I get 0=0 or 5=8. But how do you interpret the results. How can we say that something has infinitely many solutions? I don't get it.(2 votes)
- Think of it this way: when are those statements true? Is 5 ever equal to 8? Nope. No matter what, 5 does not equal 8, so in that situation, there are no solutions. What about 0 = 0? Well, by the reflexive property of equality, we know that this is always true. 0 always equals 0, so there are infinitely situations where this is true. Does that answer your question?(6 votes)

- Is it possible to have 2 solutions?(2 votes)
- Did you watch the video? Sal shows you that if there are two solutions, then the equations in the linear system create the same line. This means all points on the line are solutions, not just two.(6 votes)

- whickof the following is a system of lienear equations in two variables(3 votes)
- there can be either no solution, exactly one solution, or an infinite number of solutions. If you are dealing with two lines then the lines will either never intersect, intersect at only one point, or be on top of each other. If the latter occurs, there are an infinite number of solutions. If they only intersect at one point, then the coordinates of that point is the solutions. If there is no solution, then the lines will never intersect.(3 votes)

- If I come up with zero as the denominator of an equation, would that system have an infinite number of solutions or no solution? (I truly wonder, because it seemed that whichever answer I put in those cases the other was deemed correct.)(2 votes)
- Division by 0 is undefined. If you encounter division by 0, then whatever you were doing that resulted in division by 0 has no mathematical meaning. You generally cannot draw any conclusions if you have division by 0.

With linear equations, there are only three possibilities:

There are zero solutions.

There is one solution.

There are infinitely many solutions.

Thus, anytime you know there is more than one solution, you instantly know there are infinitely many solutions.

NOTE: This only applies to straight lines. If you have any other kind of function, the rules for how many solutions there can be are different.(5 votes)

- What if it weren't linear?(4 votes)
- Is it really just the three?

1) no solution

2) one solution

3) infinite solution

Is there no possible occurrence that there may be more than one but not infinite?(2 votes)- The solutions to the system of equations represent the point(s) that are solutions to both equations. Or, the point(s) they have in common. Linear equations create straight lines. As such, there are only the 3 scenarios.

1) No solution: This occurs when the 2 lines are parallel. Since they never tough, they have no points in common.

2) One solution: This occurs when the two lines intersect. The solution is the one point where the lines cross over each other.

3) Infinite solutions: This occurs when the two equations create the same line, so all points are in common.

To have more than one solution and not have infinite solutions, one or both of the equations needs to create a graph that curves. If their graph curves, then they are not linear equations.

Hope this helps.(4 votes)

## Video transcript

You are solving a system
of two linear equations in two variables. You have found more
than one solution that satisfies the system. Which of the following
statements is true? So before even reading
these statements, let's just think
about what's going on. So let me draw my axes here. Let's draw my axes. So this is going to
be my vertical axis. That could be one
of the variables. And then this is
my horizontal axis. That's one of the
other variables. And maybe, for sake of
convention, this could be x, and this could be y, but they're
whatever our two variables are. So it's a system of
two linear equations. So if we're graphing them,
each of the linear equations in two variables can be
represented by a line. Now, there's only
three scenarios here. One scenario is where the
lines don't intersect at all. So the only way
that you're going to have two lines in two
dimensions that don't intersect is if they have the
same slope and they have different y-intercepts. So that's one
scenario, but that's not the scenario that's
being described here. They say, you have found
more than one solution that satisfies the system. Here there are no solutions. So that's not the scenario
that we're talking about. There's another
scenario where they intersect in exactly one place. So they intersect in
exactly one place. There's one point, one
xy-coordinate right over there that satisfies both
of these constraints, but this also is not the
scenario they're talking about. They're telling us that you have
found more than one solution that satisfies the system. So this isn't the
scenario either. So the only other scenario
that we can have-- we don't have parallel lines
that don't intersect. We don't have lines that
only intersect in one place. The only other scenario is that
we're dealing with a situation where both linear
equations are essentially the same constraint. They both are
essentially representing the same xy-relationship. That's the only way that
I can have two lines, and this only applies to
linear relationship and lines. But the only way that
two lines can intersect more than one place is if
they intersect everywhere. So in this situation,
we know that we must have an infinite
number of solutions. So which of these
choices say that? This one right here-- "there are
infinitely many more solutions to the system"--
right over there.