Proportion

This is an AoPSWiki Word of the Week for Sep 13-19

Two numbers are said to be in proportion to each other if some numeric relationship exists between them. There are several types of proportions, each defined by a separate class of function.

Direct proportion

Direct proportion is a proportion in which one number is a multiple of the other. Direct proportion between two numbers x and y can be expressed as:

where k is some real number.

The graph of a direct proportion is always linear.

Often, this will be written as .

Inverse proportion

Inverse proportion is a proportion in which as one number's absolute value increases, the other's decreases in a directly proportional amount. It can be expressed as:

where k is some real number that does not equal zero.

The graph of an inverse proportion is always a hyperbola, with asymptotes at the x and y axes.

Exponential proportion

A proportion in which one number is equal to a constant raised to the power of the other, or the logarithm of the other, is called an exponential proportion. It can be expressed as either:

or

for some real number k, where k is not zero or one.

Problems

Introductory

Suppose is the solution to the system of equations:

Find k.