feature of linear equations

gullpacha

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i read in book that linear equation have one degree variable and a little further when i go i saw the give example 10/5=2 that this is linear that make me confused that in this example the degree of equation is -1 how it can be linear? 2-my second question is that can rational and radical equation be linear like sqr(y-9)=10 is this linear if yes how we can know that this is linear because the exponent of variable is not 1 so how we can say that this is linear?how we can know that a equation is linear or not ?
thank you
 
Hi
I cannot quite follow what you wrote. In any event, a linear function is a function of the form: f(x)=mx+b. The x appears to the power of 1. A quadratic function will take the form: f(x)=ax2+bx+c. As you can see the higher power of x is 2. A linear equation means that the variable appears with power 1. This definition of linearity extends to many other mathematical objects.
 
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i read in book that linear equation have one degree variable and a little further when i go i saw the give example 10/5=2 that this is linear that make me confused that in this example the degree of equation is -1 how it can be linear? 2-my second question is that can rational and radical equation be linear like sqr(y-9)=10 is this linear if yes how we can know that this is linear because the exponent of variable is not 1 so how we can say that this is linear?how we can know that a equation is linear or not ?
thank you
Please quote exactly what was said, in context, about 10/5 = 2. This might be one piece of something that might be called linear, but you haven't shown us enough to tell.

As for the second question, if there is a square root in an equation (which I'm guessing is what you mean by "sqr"), it is not a linear equation. But after squaring both sides, it will be.
 
Please quote exactly what was said, in context, about 10/5 = 2. This might be one piece of something that might be called linear, but you haven't shown us enough to tell.

As for the second question, if there is a square root in an equation (which I'm guessing is what you mean by "sqr"), it is not a linear equation. But after squaring both sides, it will be.
sir i mean that in rational equations the power of unknown is -1 so how it can be linear because the power of unknown is -1 it is not 1 how it make linear equations?2-i mean by sqr( radical) equations the power of these equations are 1/2 so how it can be linear you said that after squaring become linear it mean that this is another form of linear equations?
 
In 10/2=5 where is the power -1?

If you have a power of 1/2 then the expression is not linear.
 
In 10/2=5 where is the power -1?

If you have a power of 1/2 then the expression is not linear.
oh sorry that is my mistake i mean that 10/x=5 i was so confused so that i wrote wrong again i am so sorry
 
Dr. Peterson wrote you that the equation you wrote (y-9)1/2=10 is not really a non linear equation. If you square it then you get y-9=100 which is linear in y.
 
This equation is linear, it is equivalent to x=2.
No, it is NOT a linear equation. It can be transformed into a linear equation, but that changes its domain (in the original, x can't be 0).

@gullpacha, please, as I asked, show us exactly what was said about these equations, either as an image or in text (which we can translate if needed). You may be misinterpreting what they say, or they may just be wrong.

I have seen textbooks that give problems like this in a section on linear equations, because the same technique of "clearing fraction" applies; but they should not explicitly say it is linear. And an equation with the variable in a radical absolutely is not linear, though it too can be transformed into one.
 
No, it is NOT a linear equation. It can be transformed into a linear equation, but that changes its domain (in the original, x can't be 0).

@gullpacha, please, as I asked, show us exactly what was said about these equations, either as an image or in text (which we can translate if needed). You may be misinterpreting what they say, or they may just be wrong.

I have seen textbooks that give problems like this in a section on linear equations, because the same technique of "clearing fraction" applies; but they should not explicitly say it is linear. And an equation with the variable in a radical absolutely is not linear, though it too can be transformed into one.

No, it is NOT a linear equation. It can be transformed into a linear equation, but that changes its domain (in the original, x can't be 0).

@gullpacha, please, as I asked, show us exactly what was said about these equations, either as an image or in text (which we can translate if needed). You may be misinterpreting what they say, or they may just be wrong.

I have seen textbooks that give problems like this in a section on linear equations, because the same technique of "clearing fraction" applies; but they should not explicitly say it is linear. And an equation with the variable in a radical absolutely is not linear, though it too can be transformed into one.
I see your point that there is a problem with the domain of the equation. In any event you don't invoke any particular technique from non linear equations to solve this particular equation. So it is like a "fake" non linear equation.
 
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The trouble here is that "linear equation" (in one variable) is commonly defined as "an equation that can be put in the form ax + b = 0", without stating what methods are allowed to put it into that form. So, for example, I find this page, https://tutorial.math.lamar.edu/classes/alg/solvelineareqns.aspx, including rational equations that can be turned into linear equations, though without experience one would never know whether they are linear or quadratic or worse. They do this because the techniques used in solving them are (in the end) the same. (But there are no examples here with radicals, which have an even bigger issue with domain.)

On the other hand, it is also common to define "linear" as involving no powers of the variable other than the first, and these equations clearly do not fit that definition, which is the cause of the OP. Having two contradictory "definitions" is a problem.

So there is, admittedly, room for disagreement on this. But I think the differences are significant enough that it is very misleading to label such equations as linear. I personally resolve it by restricting "put in the form" to multiplying and dividing by constants, and adding or subtracting constants or variable expressions. Then I mention that some equations that are not themselves linear can be transformed into that form and subsequently solved by linear methods.
 
I just performed a small survey with a few colleagues and got both answers. But I think you wrote very clear in your last message what the issue is. After your "NO" regarding the equation not being linear I thought about the definition of linear operators, and thinking of L(x)=(x-9)1/2-10 it is clear that it wouldn't look linear to any one. In the other example : 10/x=5, x=0 is not a solution so we shouldn't care that much about it. But from a purist point of view x=0 indeed does not belong to the domain of the equation. I apologize if it sounds like a debate in etymology
 
i read in book that linear equation have one degree variable and a little further when i go i saw the give example 10/5=2 that this is linear that make me confused that in this example the degree of equation is -1 how it can be linear?
??There is no variable so it doesn't have a degree! Unless you actually mean the equation \(\displaystyle \frac{10}{x}= 2\). That has degree -1 but multiplying both sides by x gives \(\displaystyle 10= 2x\) which is linear.

2-my second question is that can rational and radical equation be linear like sqr(y-9)=10 is this linear if yes how we can know that this is linear because the exponent of variable is not 1 so how we can say that this is linear?how we can know that a equation is linear or not ?
thank you
We don't usually talk about the "degree" of a non-polynomial function. However, squaring both sides of \(\displaystyle \sqrt{y- 9}= 10\) gives \(\displaystyle y- 9= 100\) which is linear.
 
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