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 August 11th, 2018, 11:53 AM #1 Newbie   Joined: Jun 2018 From: Jordan Posts: 14 Thanks: 0 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 03:33 AM.
August 11th, 2018, 12:22 PM   #2
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 Originally Posted by Integraluser 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 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 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 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.

Last edited by skipjack; August 15th, 2018 at 03:38 AM.

 August 14th, 2018, 01:03 AM #3 Newbie   Joined: Jun 2018 From: Jordan Posts: 14 Thanks: 0 Thanks!
August 14th, 2018, 04:31 AM   #4
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Quote:
 Originally Posted by Integraluser 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...

August 15th, 2018, 01:39 AM   #5
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 Originally Posted by Denis 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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