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August 11th, 2018, 12:53 PM   #1
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Some questions related to variables and operations

Hi.

I got several algebraic questions and I don't really want them to remain a gap for me.

First one is about substituting variables / expressions;

Say that we have $\displaystyle (xy)^3=4x$. If I decide to substitute the expression $\displaystyle xy$ with the variable $\displaystyle n$, which is the new correct form of the utter expression?

1- $\displaystyle n^3 = 4x $
2- $\displaystyle n^3 = 4n/y$ (assuming $\displaystyle x = n/y$)
3- Both

The next question is, If I substitute a variable with an expression, do I have to do it everywhere? For example;
Say that $\displaystyle y = 3x$
and $\displaystyle g * \sqrt{y} = 2y$

Can it be $\displaystyle g*\sqrt{y} = 6x$ ?


Another question ; If I were to multiply / divide a fraction by a value that has many forms, can I use different forms for certain purposes?
Say I want to multiply $\displaystyle \frac{x}{y}$ with 1, but also $\displaystyle \sin^2(x) + \cos^2(y) = 1$.

is this correct?

$\displaystyle
\frac{x}{y} = \frac {x(\sin^2(x)+\cos^2(x))}{y}
$


Last question ; is the above case possible when it comes to equations too?

For example, assume that $\displaystyle g+c = 3 $

and I have the equation:

$\displaystyle
3x^2 = y
$

If I multiply both sides by 3, but in 2 different forms

$\displaystyle

9x^2 = y(g+c)

$
is this correct? ^

Thanks in advance.

Last edited by skipjack; August 15th, 2018 at 04:33 AM.
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August 11th, 2018, 01:22 PM   #2
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Quote:
Originally Posted by Integraluser View Post
Hi.

I got several algebraic questions and I don't really want them to remain a gap for me.

First one is about substituting variables / expressions;

Say that we have $\displaystyle (xy)^3=4x$. If I decide to substitute the expression $\displaystyle xy$ with the variable $\displaystyle n$, which is the new correct form of the utter expression?

1- $\displaystyle n^3 = 4x $
2- $\displaystyle n^3 = 4n/y$ (assuming $\displaystyle x = n/y$)
3- Both
both, except for the fact that (2) implies explicitly $y \neq 0$ whereas that's only implicitly stated in (1)

Quote:
Originally Posted by Integraluser View Post
The next question is, If I substitute a variable with an expression, do I have to do it everywhere? For example;
Say that $\displaystyle y = 3x$
and $\displaystyle g * \sqrt{y} = 2y$

Can it be $\displaystyle g*\sqrt{y} = 6x$ ?
have to? no.
Can you? Sure, equality is equality. If two elements are equal, you can freely substitute one for another.

Quote:
Originally Posted by Integraluser View Post
Another question ; If I were to multiply / divide a fraction by a value that has many forms, can I use different forms for certain purposes?
Say I want to multiply $\displaystyle \frac{x}{y}$ with 1, but also $\displaystyle \sin^2(x) + \cos^2(y) = 1$.

is this correct?

$\displaystyle
\frac{x}{y} = \frac {x(\sin^2(x)+\cos^2(x))}{y}
$
Assuming you mean $\sin^2(x)+\cos^2(x)=1$ then yes. You can multiply an expression by $1$ or by any other expression that equals $1$ and retain the equality.


Quote:
Originally Posted by Integraluser View Post

Last question ; is the above case possible when it comes to equations too?

For example, assume that $\displaystyle g+c = 3 $

and I have the equation:

$\displaystyle
3x^2 = y
$

If I multiply both sides by 3, but in 2 different forms

$\displaystyle

9x^2 = y(g+c)

$
is this correct? ^

Thanks in advance.
Yes, that's valid. Equality is equality.
Thanks from topsquark and Integraluser

Last edited by skipjack; August 15th, 2018 at 04:38 AM.
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August 14th, 2018, 02:03 AM   #3
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Thanks!
Integraluser is offline  
August 14th, 2018, 05:31 AM   #4
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Quote:
Originally Posted by Integraluser View Post
First one is about substituting variables / expressions;

Say that we have $\displaystyle (xy)^3=4x$. If I decide to substitute the expression $\displaystyle xy$ with the variable $\displaystyle n$, which is the new correct form of the utter expression?

1- $\displaystyle n^3 = 4x $
2- $\displaystyle n^3 = 4n/y$ (assuming $\displaystyle x = n/y$)
3- Both
Well, you can assign values to x and y, then check results yourself...
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August 15th, 2018, 02:39 AM   #5
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Quote:
Originally Posted by Denis View Post
Well, you can assign values to x and y, then check results yourself...
Yep I know that it would give the same answer but my question wasn't whether it would be the same in terms of value or not. All I wanted to know is whether is it permissible to do such operations or not. (For e.g: having a new variable in terms of 2 sub-variables and then using it along with another sub-variable in the same expression such as my first example)
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