- 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
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.
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- 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?(23 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.(12 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(20 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.(8 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!(8 votes)
- Can -7x+3=2x+2-9x equal to 1=0?
Then I am pretty sure the -7x's cancel out so:
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)
- 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)
What does Sal mean by "there is no number/answer that can make three become five"(or when he said that there is no way to make 3 equal to 2??) at about0:50?? i mean if you do 3 times 1.66666666667? i know it's complicated but wouldn't that be an option??
i am so and still confused(even after watching the video twice!):'((((
someone plz help explain this! T ^ T(3 votes)
- There is no variable left in the equation. It just says 3=2 which is a false statement. The number 3 will never equal 2.
In order to take your approach, you would need a variable to insert you selected value into the equation. If that had been the case, you would have an equation with one solution.
5=3 No solution because the number 5 will never equal the number 3
5=3x you can solve for X and get a solution of x =5/3
Hope this helps you to see the difference.(4 votes)
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.