mike_flower
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Balancing an equation
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lev888
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Is it called balancing? Seems "balancing an equation" is used in chemistry. I would say "solving for n".
Isolating a Variable
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Dr.Peterson
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The equation appears to be [MATH]N = \frac{P - 110}{3} + P^n[/MATH]. If so, the variable "n" (lower case) appears only once, and you can use logs to solve for it.
On the other hand, if you did not intend to make a case distinction and the equation is really [MATH]n = \frac{P - 110}{3} + P^n[/MATH], then it can't be solved algebraically, because n appears both inside and outside of an exponent. You'd have to solve numerically.
In the former case, please make an attempt and show us your work, so we can see what help you need. The first step (since you asked how to get started) will be a subtraction.
On the other hand, if you did not intend to make a case distinction and the equation is really [MATH]n = \frac{P - 110}{3} + P^n[/MATH], then it can't be solved algebraically, because n appears both inside and outside of an exponent. You'd have to solve numerically.
In the former case, please make an attempt and show us your work, so we can see what help you need. The first step (since you asked how to get started) will be a subtraction.
I think what was meant was maintaining equality while isolating the variable, which does involve the process of making balancing or equivalent changes to both sides of the equation. (I frequently explain the process in terms of an analogy to a physical balance.)
The problem is that, depending on how you interpret the problem, either there is no algebraic solution, or else the key balancing comes in the form of applying the same function to both sides of the equation, which sort of goes beyond the analogy of what you can do with a physical balance
Alternatively, multiply both sides by 3 to clear fractions.
lev888
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I understand how the term makes sense in chemistry - knowing which molecules are on the left and right sides we balance the equation by calculating the number of molecules so that the number of atoms is the same.
But in math the equation is already balanced by definition.
Valid equations are balanced by definition. Changes to a valid equation are not balanced by definition. One must ensure that the changes to each side of a valid equation are balanced in order to ensure that the resulting equation remains valid.
I am not advocating the usage of "balance the equation" to mean "isolate the variable." I am merely saying that students use it, and the rationale for its use arises out of a valid and commonly used analogy.
Dr.Peterson
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My first thought about the use of "balance" in the question was to wonder if it might be related to the Arabic terms used for algebra (e.g. if the question comes from a culture close to Arabic). For example, in https://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Khwarizmi.html I find:
Of course, here it doesn't refer to the general process of solving, but only to the idea of adding [actually, subtracting] the same thing on both sides; and therefore would be a misnomer if used in the question to mean "solve for n".
The reduction is carried out using the two operations of al-jabr and al-muqabala. Here "al-jabr" means "completion" and is the process of removing negative terms from an equation. For example, using one of al-Khwarizmi's own examples, "al-jabr" transforms x2 = 40 x - 4 x2 into 5 x2 = 40 x. The term "al-muqabala" means "balancing" and is the process of reducing positive terms of the same power when they occur on both sides of an equation. For example, two applications of "al-muqabala" reduces 50 + 3 x + x2 = 29 + 10 x to 21 + x[MATH]2[/MATH] = 7 x (one application to deal with the numbers and a second to deal with the roots).
Of course, here it doesn't refer to the general process of solving, but only to the idea of adding [actually, subtracting] the same thing on both sides; and therefore would be a misnomer if used in the question to mean "solve for n".