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### Course: Algebra 1>Unit 2

Lesson 3: Analyzing the number of solutions to linear equations

# 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?
• 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.
• Though I might have been overlooking something...

I have a question about the second scenario, -7x+3=2x+2-9x. Sal adds 7x to the both sides first, but is this acceptable to add/subtract constants (idk how should I call these numbers) first? The result is 0=-1 in this case and there are also no solutions, still I want to make it clear whether there is an order of calculation or not.
• I was kind of wondering the same thing. Me and my sister have found Khan Acadamy's math courses rather confusing, while their videos normally make sense, a lot of their exercises are really confusing.
• 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.
• 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
• 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.
• if x=5 than wound it have infinite solutions because 5=5
• 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.