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

Lesson 9: 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

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# Creating an equation with infinitely many solutions

Sal shows how to complete the equation 4(x - 2) + x = 5x + __ so that it has infinitely many solutions. Created by Sal Khan.

## Want to join the conversation?

- is it possible for an equation to have more than one solution but not infinite?(23 votes)
- It is possible for other equations (e.g., quadratic equations) to have more than one solution, but in terms of linear, no.(21 votes)

- Is there a faster way to find out what math problems are infinite solutions? I do the problems and then see if they are infinite, but it takes a long time...(7 votes)
- I simplify each side of the equation. If they are the same, then you've got infinitely many solutions.

Here is an example:

23x-5-12x=11(x-1)-6 is equal to 11x-5=11x-5

Hope this helps. God bless!(12 votes)

- is anyone else finding this too be incredibly difficult? like I have watched this video numerous times but the subject is still extremely difficult and foreign for my brain to wrap around. am I stupid or is it a difficult subject?(9 votes)
- Creating problems is a much hinger level thinking process than just solving. In this case, the idea is that you have to create something that makes both the right side of the equation and the left side to be equal to each other which gives you an infinite number of solutions. so if you have 5x-8 on the left, you need 5x-8 on the right for everything to cancel and end up with 0=0.

What do you think would be the answer to the "?" in 3x + 6 = 3(x + ?) to have infinite solutions?(4 votes)

- Why do we need to know how many solutions there are to each equation??

P.S. I’m not meaning to be sarcastic or rude, I’m genuinely asking!(3 votes)- When you are asked to solve an equation, you are being asked to find all values that will make the equation be true. Equations with one variable that are linear equation have 3 possible solution scenarios.

1) The variable has one solution

2) The equation is a contradiction (always false), so it has no solutions.

3) The equation is an identity (always true), so the variable has a solution set of all real numbers. In other words, any number you can imagine will make the equation be true. In this scenario, there are infinite solutions.

Understand the number of solutions helps you to identify what is the solution set to the equation.(8 votes)

- when you multiply negative and positive numbers what answer do you get?(4 votes)
- You get a negative number- negative*positive=negative.

For an example: -2*3=-6.(4 votes)

- I don't get what he says at0:49because if 4x-8+x wouldn't it be 4x minus a positive x because if we remove the 8 then it would be 4x-x and that would give us 3x. I need help. I have a math test about equations please help!(3 votes)
- It is 4x-8+x. This can be rearranged to 4x+x-8. You subtract the 8 no matter how you rearrange the expression, not the x. You add the x. The signs of variables and/or numbers in an expression never change, no matter how you rearrange them.(3 votes)

- I need to use the equation 2x-3y=6 to write an equation that will create a linear system with infinite many solutions,can someone help me(2 votes)
- A linear system with infinite solutions has 2 equations that end up being the same line. You could change your equation into slope-intercept form. It would be the same equation, but look different. Another way, you could multiply or divide your equation by a constant and you have an equivalent equation.

Any system that has your given equation and an equivalent equation would be a system with infinite solutions.(6 votes)

- Why so many questions?(4 votes)
- I am working multistep equations with variables on both sides and I do not understand how to work the problems and am very confused. I have problems for example like f(b)=2b+6

g(b)=b+3

Can you please show me how to work problems such as this?(3 votes) - what does Sal mean when he says "any X"?(1 vote)
- The solution to the equation is: x = all real numbers. You randomly pick any number to use for x, plug it into the equation, and the 2 sides of the equation will be equal. So, it will be a solution to the equation.

Hope this helps.(5 votes)

## Video transcript

We're asked to use the drop-down
to form a linear equation with infinitely many solutions. So an equation with
infinitely many solutions essentially has the same
thing on both sides, no matter what x you pick. So first, my brain just wants
to simplify this left-hand side a little bit and
then think about how I can engineer the
right-hand side so it's going to be
the same as the left no matter what x I pick. So right over here,
if I distribute the 4 over x minus
2, I get 4x minus 8. And then I'm adding x to that. And that's, of course, going
to be equal to 5x plus blank. And I get to pick
what my blank is. And so 4x plus x is 5x. And of course, we
still have our minus 8. And that's going to be
equal to 5x plus blank. So what could we make
that blank so this is true for any x we pick? Well, over here we have
5 times an x minus 8. Well, if we make this a minus
8, or if we subtract 8 here, or if we make this
a negative 8, this is going to be true for any x. So if we make this
a negative 8, this is going to be true
for any x you pick. You give me any x, you multiply
it by 5 and subtract 8, that's, of course, going to
be that same x multiplied by 5 and subtracting 8. And if you were to try to
somehow solve this equation, subtract 5x from both sides,
you would get negative 8 is equal to negative 8, which is
absolutely true for absolutely any x that you pick. So let's go-- let me actually
fill this in on the exercise. So I want to make 5-- it's
going to be 5x plus negative 8.