Dividing cancels out a solution?

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In summary, the conversation discusses dividing out a variable in an equation and the potential loss of solutions that may occur. It also mentions taking the square root of both sides, but this does not solve for the variable and can also result in lost solutions. The solution method of factoring is suggested as an alternative.
  • #1
ainster31
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Context:

In the video, he divides out a variable as below:
$$r^2=2rsin\theta\\r=2sin\theta$$
When you do a division like this, aren't you dividing out a variable?

What happens if you take the square root of both sides like below?
$$r^2=2rsin\theta\\|r|=\sqrt{2rsin\theta}$$
Would that be more correct?
 
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  • #2
Yes, he is. I particular, he has "lost" the r= 0 for all [itex]\theta[/itex] solution.

I'm not sure what you mean by "more correct" for [itex]|r|= \sqrt{2r sin(\theta)}[/itex]. It is not at all a "solution" because you have not "solved for r".
 
  • #3
ainster31 said:
Context:

In the video, he divides out a variable as below:
$$r^2=2rsin\theta\\r=2sin\theta$$
When you do a division like this, aren't you dividing out a variable?
Well, of course. The more important point is that you are potentially losing a solution of the equation; namely, r = 0. In this case, it doesn't matter because the equation r = 2sin(θ) still has r = 0 when θ = 0 + k##\pi##.
ainster31 said:
What happens if you take the square root of both sides like below?
$$r^2=2rsin\theta\\|r|=\sqrt{2rsin\theta}$$
Would that be more correct?
No, and besides, what's the point? You haven't solved for r since it still appears on both sides of the equation.
 
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  • #4
How did you know he cancels out the ##r=0## solution?
 
  • #5
ainster31 said:
How did you know he cancels out the ##r=0## solution?

Because that's the solution that always disappears when you divide by a variable. The only root of ##rA(...) = rB(...)## with ##A\neq{B}## is ##r=0##, and that's the one that we're losing when we divide both sides by ##r## to turn the equation into ##A(...) = B(...)##.
 
  • #6
ainster31 said:
How did you know he cancels out the ##r=0## solution?

Any time you divide both sides of an expression by something, you should check to see whether that thing can be zero. If it can be, then you have lost solutions corresponding to when that equals zero. For example, if someone asks you to solve:

(x-1)(x-2) = (x-1)(2x-1)

The first thing we do is divide both sides by x-1
(x-2) = (2x-1)

Now we solve for x
x = -1

OK, but we divided both sides by x-1, and we have to be worried about when that is equal to zero. x-1= 0 when x=1, so I should go back to the original equation and check that x=1 is a solution, which it is easily seen to be.
 
  • #7
ainster31 said:
Context:

In the video, he divides out a variable as below:
$$r^2=2rsin\theta\\r=2sin\theta$$
When you do a division like this, aren't you dividing out a variable?

What happens if you take the square root of both sides like below?
$$r^2=2rsin\theta\\|r|=\sqrt{2rsin\theta}$$
Would that be more correct?


Simply do:

r^2=2rsin\theta\ <-> r(r-2sin\theta\)=0 <-> r = 0 or r = 2sin\theta\


problem solved !

(sorry I don't know how to do the maths letters here )
 
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1. What does it mean to "divide cancels out a solution"?

Dividing cancels out a solution refers to the process of dividing both sides of an equation by the same number or variable. This allows you to simplify the equation and find the value of the variable without changing the solution.

2. Why is it important to divide cancels out a solution?

Dividing cancels out a solution is important because it allows you to solve equations and find the value of the variable. It is also a fundamental step in algebraic manipulation and can help simplify complex equations.

3. Can dividing cancels out a solution be applied to all types of equations?

Yes, dividing cancels out a solution can be applied to all types of equations, including linear, quadratic, and exponential equations. However, it is important to note that in some cases, the solution may not be valid for the original equation.

4. What are the common mistakes when dividing cancels out a solution?

One common mistake when dividing cancels out a solution is dividing by zero, which is undefined. It is also important to be careful when dealing with fractions and to ensure that the same number or variable is being divided on both sides of the equation.

5. Are there any alternative methods to dividing cancels out a solution?

Yes, there are alternative methods to dividing cancels out a solution, such as using the multiplication property of equality or combining like terms. However, dividing is often the most straightforward and efficient method for solving equations.

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