How do you write an equation that describes the relationship between given variables?

Answer 1

Examples are discussing the relationship between the surface area and volume of a rectangular prism to its height, length and width. For geometric examples, it's usually a good idea to draw the object and label what you call each thing (especially in 2d, since any 2 of base, height, length and width can be used to describe most objects)

Step 1: define your symbols. If you're in a class, do this even if it's dead obvious; it's good practice for when things get a lot more complex (viscosity, for instance is denoted by the greek letter 'mu' in fluid dynamics), plus math teachers tend to give marks (or take marks off) based on the presence/absence of each step

ie:

W=width

L=length

H=height

A=surface area

V=volume

Step 2: Determine how the input and output parameters are related.

ie:

surface area=sum of area on faces

2 faces are width by length, 2 are length by height and 2 are width by height

volume is base area times height; base area is area of a face not including height, therefore area of the width by length face.

Step 3: Convert relationships into symbolic form

2 faces of width by length=2WL

2 faces of width by height=2WH

2 faces of length by height=2LH

Base area=width by length area => B=WL

Volume = base area times height => V=BH

Since surface area is the sum of all face areas:

A= 2WL + 2WH + 2LH

And since we want to define volume in terms of input parameters rather than intermediates:

V= WLH

In algebra classes, you should be given the relationship in non-equation form if you're asked to do this, or it should be something quite simple like the example I used.

There are methods for determining the relationship between variables if you only have a data table, but that's Calculus, not Algebra.

There is also a method called Pi Notation for determining a relationship between parameters based on what is being measured (for instance, one cannot determine the amount of light being generated by an object from knowing its dimensions, unless one can determine a conversion factor that tells you how much light is generated by some dimensional parameter (unit of surface area or volume), but that's even further beyond this than Calculus; I learned it in second year (university) fluid dynamics. However, the basic principles aren't that complex - if you're interested in science, I suggest checking it out, since it's a very useful tool in scientific work.

Answer 2

y = 4x

Answer 3

h =l4