A popular method that is used in the colleges is called the "algebraic" method and simply involves an algebraic relationship between two or more variables with one variable expressed as a function of one of the others. For example, Density is a function of mass and volume, this is stated generally as D = f(m and V). The precise definition of this relationship is called the algebraic expression.

Suppose we set out to investigate this relationship between density, mass and volume and in order to precisely define this relationship, we do an investigation in a laboratory setting on the relationship between the density of several substances manfested as different objects and the corresponding mass of these objects. If we take several objects made of different substances, and we hold their shapes and sizes the same (ie: their volumes), we would find in our investigation that holding the volume of several objects made of different substances constant, then the mass of the different objects will differ in relation to their densities. If the mass of one object in the set is larger than another object then we would observe that the density of the first object will be correspondingly greater than the second. Mathematically, we can conclude from our study that density is directly related and linear to mass provided all other related variables (ie volume) remain constant and fixed in value. Notice that I am using the term linear here. Not all direct relationships are linear, but these will be. Linearity has to do with the one on one increase or decrease of one variable related to another.

For example, the density and mass relationship is a direct relationship in that if one observes one variable between one of the objects in the set of objects being larger in size or value then another object in that set, then one would observe the size or value of the other variable in the first object to be correspondingly larger than the second object assuming all other related variables remain constant during the comparison. However, if I said that if one object has double the value of mass as another object then this leads to an observation that the density of the first object is double the density of the second object ,or if one observes that the mass of one object is half the mass of another object in the set then this leads to the density of the first object being half the density of the second object. If we compare one object that has three times the mass of another object in the set then one finds that the density of the first object is triple the density of the second. What we are, in essence, saying is that the change observed in one variable corresponds to the exact same change in the SAME PROPORTION that we observe in the other variable. THAT is a linear relationship.

A linear relationship can be depicted as an equality between two ratios so the so called ratio and proportion method used so commonly in the secondary schools only applies to linear relationships. That severely restricts its use in the physical sciences , and is the reason ratio and proportion method of solving problems is usually not considered at the college level. In a linear relationship. the increment of change by one variable is met by the same increment of change by the other variable. If I have two objects of the same volume and one object has four times the mass of the other object, then we can accurately conclude that the first object will have a density four times greater than the second object.

On the other hand, if we compare the volume of two equally massive objects in a set of objects to their corresponding densities , then we will find out through experimentation that those objects that are lower in volume at constant mass will have correspondingly higher densities. Mathematically, we say that density is "inversely" related to the volume provided the mass remains constant or fixed in value.

What does being inversely related mean? If we notice a change in the volume of two of our objects by a factor of two (ie: double it) while having the masses of the two compared objects remain fixed, then we find that the density of objects have a density of 1/2 If we were to change the mass of an object while holding the density constant what would be the relationship between the mass and the volume? The mass of each of these objects could be measured on a mass balance. Comparing the mass of each object to its corresponding volume would lead one to conclude that the smaller massed objects of this substance would have correspondingly smaller volumes associated with them. In fact, one would observe that if one object is half as massive as another object, then its volume would be half as great. If one object is twice as great in mass as another object then its volume would be twice as great. This should sound familiar. This sounds like mass is linearally related to volume provided the density remains constant. We can use this relationship between the mass and the volume of a substance to derive or develop an algebraic expression mathematically defining the relationship that exists between these two variables, mass and volume. This can be done using a formula known as the slope Intercept (straight line) formula. The development of this algebraic expression and the derivation of the straight line formula itself is a topic for another lesson.

However the end result is that Density = Mass / volume

Algebraic Manipulation (Transformation of Variables)

Algebraic manipulation of variables in a math formula involves a process referred to as "transformation". When we use the algebraic solution to solve a math problem, we identify which variables afre known and which variable is being requested or looked for. You isolate the unknown variable in the formula on one side and all other variables and constants are on the other side of the equality.

How do we move all but the unknown variable on the other side of the equality?

This is accomplished by transforming the variables and constants to the other side. This can be done by performing the OPPOSITE math operation indicated. However when you perform the opposite mathematical operation on one side it must be done on the other side. If not the statement of equality will be upset. Let's take an example. Suppose we wish to solve the following formula for the variable "C":

F = 1.8C + 32

1. We wish to isolate the variable C

2. To do this we first move (transform) the constant 32 to the other side by performing the opposite math operation then what is indicated which is "addition". So to move the 32 to the other side, we will subtract 32 on the right, but we must also subtract 32 from the other side:

   F - 32 = 1.8C + 32 - 32

   Now the equation looks like:

   F - 32 = 1.8C

3. But the constant 1.8 must be moved to the other side. To accomplish this we must perform the opposite math operation to multiplication which is division. We must divide BOTH sides by 1.8. If we do this we will have the following:

   (F - 32) / 1.8 = 1.8C/1.8

   or

   (F-32) / 1.8 = C

We have now solved the above equation for the variable "C".

Keep in mind that the opposite math operation of multiplication is division and the opposite math operation to addition is subtraction.

Here are some problems for you to try in algebraic manipulation:

1. Solve for the variable n in the following equation:

   PV = nRT

2. Solve for T1 in the following equation:

   V₁ / T₁ = V₂ / T₂

3. Solve for F in the following:

   C = 1.8F + 32

4. If variable A is inversely related to variable B and A is linearly related to C, then express an algebraic equation that would reflect these relationships.

5. Solve for P in the following:

   PV = nRT

   Click here if you wish to check your answers to these problems.

   The Continuity Principle of the Universe

   All of this comparison results is predicated on an important universal assumption, the continuity of the universe. If I, for example, use specific objects made of specific substances in the above investigation, I am assuming that using some other different substance for the objects will lead to the same conclusions. That is called the uniformity or continuity principle of the universe, and is absolutely essential for any deductive or inductive reasoning.

   Empirical vs intuitive mathematical formulas

   Most algebraic expressions defining the physical relationships between two or more variables are developed either in an experimental or investigatory manner or they are developed intuitively. When developed in an experimental setting the resulting formulas are referred to as "empirical" mathematical formulas. The word empirical means literally "based on experimentation". Experimentation involves the controlling of variables within the experimental design. Some variables are held constant while others are allowed to vary. When investigating the relationship between two variables with other defined variables present in the same physical environment, the other variables must be held by the experimenter fixed so as to get a precise definition on the relationship between the two variables under investigation. Otherwise, one could never quite know if any changes between the values of two variables was due to their relationship toward one another or the relationship of some other variable that was not held fixed during the investigation. When making comparisons as we did earlier in this discussion, the same procedure must be observed. We had to hold one of the three variables constant in order to get a definition of the relationship of the other two variables to one another.

   Independent vs dependent variables

   In the above setting, we were doing what is called a "comparative analysis" where we took objects as they were and compared them in order to make conclusions about relationships between two variables. Of course, in the laboratory, we alter one of the two variables that are being investigated in order to see how the other variable will respond to the change manipulated by the experimenter. The variable that is being pre-meditatively altered by the experimenter is called the "independent variable". The variable that responds by changing its value as a result of the independent variable being manipulated is called the "dependent variable". For example in the investigation of the relationship between the volume of a constant amount of a gas sample and its pressure, we were to design an experimental setup or apparatus by which we might manipulate the pressure then the pressure variable would be called the independent variable. We would then be able to measure the resulting value for the volume variable. The volume in this case would be defined as the dependent variable. Its value will depend upon what we set the pressure value to be.

   This concludes the algebraic Method lesson. If you have any comments or questions, feel free to contact the author by clicking on the mailbox icon below.

   R. H. Logan, Instructor of Chemistry, Dallas County Community College District, North Lake College.
   

   ---

   Acknowledgements

   Send Comments to R.H. Logan: rhlogan@ix.netcom.com

   All contents copyrighted (c) 1995
   R.H. Logan, Instructor of Chemistry,DCCCD
   All Rights reserved
   
   Revised: 11/8/97
   
   
   Return to Basic Math menu 
   
   
   
   Return to Home Page
   
   
   
   
   Solutions To Algebraic Manipulation Homeework Set
   
   
   Solve for the variable n in the following equation:
   
   PV = nRT
   
   n = PV / RT
   
    
   Solve for T1 in the following equation:
   
   V1 / T1 = V2 / T2
   
   
   V1T2 / V2 = T1
   
   Solve for F in the following:
   
    X  = 1.8F + 32
   
   
   F = X - 32 / 1.8
   
   If variable A is inversely related to variable B and A is linearly related to C, then express an algebraic equation that would reflect these relationships.
   
   A = C / B
   
   Solve for P in the following:
   
   PV = nRT
   
   P = nRT / V
   
   Return to the lesson.
   

URL:http://members.aol.com/profchm/algebra.html