Main content

## Algebra 1

### Course: Algebra 1 > Unit 2

Lesson 3: Analyzing the number of solutions to linear equations- Number of solutions to equations
- Worked example: number of solutions to equations
- Number of solutions to equations
- Creating an equation with no solutions
- Creating an equation with infinitely many solutions
- Number of solutions to equations challenge

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Number of solutions to equations

A linear equation could have exactly 1, 0, or infinite solutions. If we can solve the equation and get something like x=b where b is a specific number, then we have one solution. If we end up with a statement that's always false, like 3=5, then there's no solution. If we end up with a statement that's always true, like 5=5, then there are infinite solutions.. Created by Sal Khan.

## Want to join the conversation?

- Why is it that when the equation works out to be 13=13, 5=5 (or anything else in that pattern) we say that there is an infinite number of solutions?(21 votes)
- 13=13 Is a true statement...that is why.(3 votes)

- Does the same logic work for two variable equations? Is there any video which explains how to find the amount of solutions to two variable equations? Help would be much appreciated and I wish everyone a great day!(10 votes)
- For a system of two linear equations and two variables, there can be no solution, exactly one solution, or infinitely many solutions (just like for one linear equation in one variable).

If the two equations are in standard form (both variables on one side and a constant on the other side), then the following are true:

1) lf the ratio of the coefficients on the x’s is unequal to the ratio of the coefficients on the y’s (in the same order), then there is exactly one solution.

2) lf the coefficients ratios mentioned in 1) are equal, but the ratio of the constant terms is unequal to the coefficient ratios, then there is no solution.

3) lf the coefficient ratios mentioned in 1) and the ratio of the constant terms are all equal, then there are infinitely many solutions.(10 votes)

- At3:09, in the first example, why did he subtract x? Weren't they already equal, or did I miss something?(3 votes)
- Don’t worry, you didn’t miss anything. :)

Let’s review the idea of ”number of solutions to equations” real quick. Basically, an equation can have:**Exactly one solution**, like 2x = 6. It solves as x = 3, no other options.**No solutions**, like x+6 = x+9. This would simplify to 6 = 9, which is, ummm, not true, so no solutions.**Infinitely many solutions**, such as 3x = 3x. This simplifies to x = x.

So there’s the part you’re likely confused about. Why does x = x mean infinitely many solutions? Well, because…**anything is equal to itself**(duh) so literally any number could be an answer. Solve x as 473? 473 = 473, yup! And 64 = 64, and -1.24 = -1.24.

Sal takes away both X’s that’s what you do when solving an equation, you do the same thing to both sides. So x = x becomes just an equal sign!

Essentially,**if you can simplify an equation down to just an equals sign, it has infinitely many solutions**.

I hope this helped! :D(16 votes)

- You know, Math makes no sense, you can literally end up with answers like this: 8=3. or something confusing like that. So why does this work?(4 votes)
- If you have ended with an expression like 8 = 3, there is an error in your solution or, if you are working with a system of equations, then there is no solution that satisfies all the equations in the system.

8 = 3 is not an answer. It either means that you need to review your work or that there is no answer.(7 votes)

- I don't know if its dumb to ask this, but is sal a teacher?(3 votes)
- Sorry, repost as I posted my first answer in the wrong box.

According to a Wikipedia page about him, Sal is:

"[a]n American educator and the founder of Khan Academy, a free online education platform and an organization with which he has produced over 6,500 video lessons teaching a wide spectrum of academic subjects, originally focusing on mathematics and sciences."

So technically, he is a teacher, but maybe not a conventional classroom one.

Hope that helped!(7 votes)

- Can -7x+3=2x+2-9x equal to 1=0?

-7x+3=2x-9x+2

-7x+3-2=-7x

Then I am pretty sure the -7x's cancel out so:

1=0

Is this still correct?(3 votes)- Your work is correct so far, but incomplete.

1=0 is a false statement (a contradiction). It is trying to tell you that the equation has no solution. You need to make that interpretation to say that the equation has no solution.(6 votes)

- so when 0=0, its always infinite solutions ?(3 votes)
- Yes, and you could even have 5=5 of -8=-8. If the variable has been eliminated and you have a number = itself, the equation is an identity (always true). You can use any value for the variable and the two sides of the equation will be equal.(5 votes)

- Does anyone know if an equation can ever have three, four, or five solutions? Like, there can be two, for example "x = square root of 9" because that could mean x = -3 or x = 3. But can there be more than two?(3 votes)
- If x=sqrt(9) didn't start as a quadratic equation, then it has one solution, 3. It is asking you to use the principal root.

If you started with x^2 = 9 (a quadratic) and solved the equation, you could solve by factoring, square root method, or quadratic formula and you would get 2 solutions x=3 and x=-3

Higher degree equation can have more solutions. The degree of the polynomial tells you the maximum number of possible solutions.

This current lesson is about linear equations with one variable. They will have one solution, no solution (if the equation turns out to be a contradiction) or a solution of all real number (if the equation turns out to be an identity). You learn about quadratic equations and higher degree polynomial equations as your progress further into algebra.(5 votes)

- if x=5 than wound it have infinite solutions because 5=5(3 votes)
- Sal put x=5 up an example of what you would see if you solved the equation and it has one solution. He didn't give you the actual equation, so you can't do the check. If you had the equation, you could verify that x=5 is a good solution. If it is, the check will show the 2 sides are equal (a number = itself). But, you would be getting this in a check, not when solving the original equation.

Hope this helps.(5 votes)

- What if you replaced the equal sign with a greater than sign, what would it look like? Would it be an infinite solution or stay as no solution(1 vote)
- Like systems of equations, system of inequalities can have zero, one, or infinite solutions. If the set of solutions includes any shaded area, then there are indeed an infinite number of solutions.(6 votes)

## Video transcript

Determine the
number of solutions for each of these
equations, and they give us three equations right over here. And before I deal with these
equations in particular, let's just remind
ourselves about when we might have one or
infinite or no solutions. You're going to
have one solution if you can, by
solving the equation, come up with something like
x is equal to some number. Let's say x is
equal to-- if I want to say the abstract--
x is equal to a. Or if we actually
were to solve it, we'd get something like x
equals 5 or 10 or negative pi-- whatever it might be. But if you could actually
solve for a specific x, then you have one solution. So this is one solution,
just like that. Now if you go and you try to
manipulate these equations in completely legitimate
ways, but you end up with something crazy
like 3 equals 5, then you have no solutions. And if you just think
about it reasonably, all of these equations
are about finding an x that satisfies this. And if you were to just
keep simplifying it, and you were to get
something like 3 equals 5, and you were to ask
yourself the question is there any x that can somehow
magically make 3 equal 5, no. No x can magically
make 3 equal 5, so there's no way that you could
make this thing be actually true, no matter
which x you pick. So if you get something
very strange like this, this means there's no solution. On the other hand, if you get
something like 5 equals 5-- and I'm just over
using the number 5. It didn't have to
be the number 5. It could be 7 or 10
or 113, whatever. And actually let
me just not use 5, just to make sure that you
don't think it's only for 5. If I just get something,
that something is equal to itself,
which is just going to be true no matter what
x you pick, any x you pick, this would be true for. Well, then you have
an infinite solutions. So with that as a
little bit of a primer, let's try to tackle
these three equations. So over here, let's see. Maybe we could subtract. If we want to get rid of this
2 here on the left hand side, we could subtract
2 from both sides. If we subtract 2
from both sides, we are going to be left
with-- on the left hand side we're going to be
left with negative 7x. And on the right
hand side, you're going to be left with 2x. This is going to
cancel minus 9x. 2x minus 9x, If we simplify
that, that's negative 7x. You get negative 7x is
equal to negative 7x. And you probably see
where this is going. This is already true
for any x that you pick. Negative 7 times that x is going
to be equal to negative 7 times that x. So we already are going
into this scenario. But you're like hey, so
I don't see 13 equals 13. Well, what if you did
something like you divide both sides by negative 7. At this point, what I'm
doing is kind of unnecessary. You already understand that
negative 7 times some number is always going to be
negative 7 times that number. But if we were to do this,
we would get x is equal to x, and then we could subtract
x from both sides. And then you would
get zero equals zero, which is true for
any x that you pick. Zero is always going
to be equal to zero. So any of these
statements are going to be true for any x you pick. So for this equation
right over here, we have an infinite
number of solutions. Let's think about this one
right over here in the middle. So once again, let's try it. I'll do it a little
bit different. I'll add this 2x and this
negative 9x right over there. So we will get negative 7x
plus 3 is equal to negative 7x. So 2x plus 9x is
negative 7x plus 2. Well, let's add-- why don't we
do that in that green color. Let's do that in
that green color. Plus 2, this is 2. Now let's add 7x to both sides. Well if you add 7x to
the left hand side, you're just going to
be left with a 3 there. And if you add 7x to
the right hand side, this is going to go
away and you're just going to be left with a 2 there. So all I did is I added 7x. I added 7x to both
sides of that equation. And now we've got
something nonsensical. I don't care what x you pick,
how magical that x might be. There's no way that that x is
going to make 3 equal to 2. So in this scenario right over
here, we have no solutions. There's no x in the universe
that can satisfy this equation. Now let's try this
third scenario. So once again, maybe we'll
subtract 3 from both sides, just to get rid of
this constant term. So we're going to get negative
7x on the left hand side. On the right hand side, we're
going to have 2x minus 1. And now we can subtract
2x from both sides. To subtract 2x from
both sides, you're going to get-- so
subtracting 2x, you're going to get negative
9x is equal to negative 1. Now you can divide both
sides by negative 9. And you are left with
x is equal to 1/9. So we're in this
scenario right over here. We very explicitly
were able to find an x, x equals 1/9, that
satisfies this equation. So this right over here
has exactly one solution.