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## Operations and Algebraic Thinking 229+

### Unit 1: Lesson 3

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.
• 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.
• 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!
• When Sal said 3 cannot be equal to 2 (at ), no matter what x you use, what if x=0? For 3x=2x and x=0, 3x0=0, and 2x0=0. This makes it true.
• If x=0,
-7(0) + 3 = -7(0) + 2
0 + 3 = 0 + 2
3 = 2
3 and 2 are not coefficients: they are constants. However, you would be correct if the equation was instead 3x = 2x.
• Guys! At I just thought of one solution to make the second equation 2=3
since 5∞=∞
then 3∞=2∞ makes sense

So we could time both sides by a number which in this equation was x, and x=infinit then this equation has one solution.
• Sorry, but it doesn't work. You are treating the equation as if it was 2x=3x (which does have a solution of 0). But, in the equation 2=3, there are no variables that you can substitute into. It is just saying that 2 equal 3. It doesn't. This is a false equation called a contradiction. It has no solution.
• is all real numbers and infinite the same thing?
• Well you could say that because infinity had real numbers and it goes forever, but real numbers is a value that represents a quantity along a continuous line.
• Makes no sense. Help!!
• All you have to do is basically solve the questions, if the numbers from both sides are equal, then it is a infinity solutions, which means that if you take a random number and put it in for the both sides where the x are, then you will get the same answer for both sides. if you do the same thing and the numbers on both sides are NOT equal, that means that there is no solution. if its x= to the number you ended up with, that means that there is only one solution. makes sense?
• how to do 2x-8=y into no solution?