xy = 5

a) Identify each relationship as direct or inverse variation.
b) Calculate the constant of each relationship and explain how you arrived at your answer.
c) Using complete sentences, explain how the two relationships are different, and how they are similar.

y = 5(1/x) is inverse

5 is the constant

You only gave me one relationship.

To identify whether the relationship between x and y is direct or inverse variation, we need to understand the definitions of these terms.

A direct variation is a relationship between two variables where an increase in one variable results in a proportional increase in the other variable. In other words, if we increase x, y will also increase. Mathematically, it can be represented as y = kx, where k is a constant.

An inverse variation is a relationship between two variables where an increase in one variable results in a proportional decrease in the other variable. In other words, if we increase x, y will decrease. Mathematically, it can be represented as xy = k, where k is a constant.

Now let's analyze the given equation: xy = 5

a) To identify the relationship as direct or inverse variation, we need to rearrange the equation into the form y = kx.

Dividing both sides of the equation by x, we get:
y = 5/x

Since y is inversely proportional to x, we can conclude that the relationship is an inverse variation.

b) To calculate the constant of the relationship, we can substitute different values for x and y and observe their relationship. Let's consider a few values:

When x = 1, y = 5/1 = 5
When x = 2, y = 5/2 = 2.5
When x = 3, y = 5/3 ≈ 1.667
When x = 4, y = 5/4 = 1.25

As we can see, different values of x result in different values of y, but their product always remains equal to 5. Therefore, the constant k is 5.

c) The two relationships, direct variation and inverse variation, are different because in direct variation, an increase in one variable leads to a proportional increase in the other variable, while in inverse variation, an increase in one variable results in a proportional decrease in the other variable.

However, they are similar in that they both involve a constant relationship between the variables. In direct variation, the constant is multiplied by x to get y, and in inverse variation, the constant is the product of x and y. In both cases, the constant remains the same for all values of x and y.

In summary, the relationship between x and y in the given equation is an inverse variation, with the constant of 5.