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## Algebra (all content)

### Course: Algebra (all content)>Unit 2

Lesson 9: Analyzing the number of solutions to linear equations

# Number of solutions to equations

See how some equations have one solution, others have no solutions, and still others have 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?
• 13=13 Is a true statement...that is why.
• 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!
• 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.
• At , in the first example, why did he subtract x? Weren't they already equal, or did I miss something?
• 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
• 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?
• 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.
• I don't know if its dumb to ask this, but is sal a teacher?
• Sorry, repost as I posted my first answer in the wrong box.

"[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!
• 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?
• 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.
• so when 0=0, its always infinite solutions ?
• 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.
• 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?
• 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.
• if x=5 than wound it have infinite solutions because 5=5