Does the relation in the table represent direct variation, inverse variation, or neither? If it is direct or inverse variation, write an equation to represent the relation. Explain your answer.

x; 5,10,15,20
y; 2,1 ,2/3,1/2

Can someone please help me with this? I really don't understand what to do.

if xy is constant, it's inverse. xy=k

If y/x is constant, it's direct. y=kx

So, take a look and see what you find.

inverse ... y gets smaller as x gets bigger

x y = 10

To determine whether the relation in the table represents direct or inverse variation, we need to check if the ratio between x and y values remains constant. Let's examine the given values:

When x = 5, y = 2
When x = 10, y = 1
When x = 15, y = 2/3
When x = 20, y = 1/2

To find the ratio between x and y, let's divide y by x for each value:

y/x; 2/5, 1/10, 2/15, 1/20

Simplifying these fractions, we have:

y/x; 2/5, 1/10, 2/15, 1/20

Since the ratio y/x is not constant, it means that the relation does not represent direct variation.

To determine if it represents inverse variation, we need to check if the product of the x and y values is constant. Let's multiply x by y for each value:

x * y; 5 * 2 = 10, 10 * 1 = 10, 15 * 2/3 = 10, 20 * 1/2 = 10

As the product of x and y is always 10, we can conclude that the relation represents inverse variation.

To write an equation to represent inverse variation, we can use the form:

xy = k

where k is the constant. In this case, the constant is 10.

So, the equation representing the relation is:

xy = 10

To determine whether the relation in the table represents direct variation, inverse variation, or neither, we need to examine the pattern between the x and y values.

Direct variation occurs when two variables are directly proportional to each other, meaning they increase or decrease at the same rate. In direct variation, if one variable doubles, the other variable also doubles. To verify if the relation is a direct variation, we need to check if the ratio between y and x remains constant.

Inverse variation occurs when two variables are inversely proportional to each other, meaning they have an opposite relationship. In inverse variation, as one variable increases, the other variable decreases, and vice versa. In this case, to verify if the relation is an inverse variation, we need to check if the product of y and x remains constant.

Let's calculate the ratios for the given values of x and y:

For x = 5, y = 2
Ratio = y / x = 2 / 5 = 0.4

For x = 10, y = 1
Ratio = y / x = 1 / 10 = 0.1

For x = 15, y = 2/3
Ratio = y / x = 2/3 / 15 = 0.0444...

For x = 20, y = 1/2
Ratio = y / x = 1/2 / 20 = 0.025

Since the ratios are not constant, the relation in the table does not represent either direct variation or inverse variation.

Therefore, we cannot write a simple equation to represent the relation in this case.

It's important to note that not all relations can be described by direct or inverse variation. Some relations may have more complex patterns or behavior that cannot be easily described using these concepts.