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 situation.
Explain your answer
x 5, 10, 15, 20
Y 2, 1, 2/3, 1/2
note that xy = 10
its a inverse variation
To determine if the relation represents direct variation, inverse variation, or neither, we need to check if the ratio of y to x remains constant.
Let's calculate the ratio of y to x for each pair of values:
For (5, 2): y ÷ x = 2 ÷ 5 = 0.4
For (10, 1): y ÷ x = 1 ÷ 10 = 0.1
For (15, 2/3): y ÷ x = (2/3) ÷ 15 = 0.0444
For (20, 1/2): y ÷ x = (1/2) ÷ 20 = 0.025
Since the ratio of y to x is not constant, it does not represent direct variation.
Next, let's calculate the product of x and y for each pair of values:
For (5, 2): x × y = 5 × 2 = 10
For (10, 1): x × y = 10 × 1 = 10
For (15, 2/3): x × y = 15 × (2/3) = 10
For (20, 1/2): x × y = 20 × (1/2) = 10
Since the product of x and y is constant (10), it represents inverse variation.
The equation representing inverse variation is:
x × y = k
where k is the constant of variation. In this case, k is 10.
To determine whether the relation in the table represents direct variation, inverse variation, or neither, we need to see if the ratio between the x-values and the y-values remains constant.
For direct variation, the ratio between x and y should be constant. In other words, if we divide any x-value by its corresponding y-value, we should always get the same result.
Let's check:
x = 5, y = 2
5/2 = 2.5
x = 10, y = 1
10/1 = 10
x = 15, y = 2/3
15/(2/3) = 22.5
x = 20, y = 1/2
20/(1/2) = 40
As we can see, the ratios are not constant for direct variation since they are all different. Therefore, we can conclude that the relation in the table does not represent direct variation.
Now let's explore inverse variation. For inverse variation, the product of x and y should be constant. In other words, if we multiply any x-value by its corresponding y-value, we should always get the same result.
Let's check:
x = 5, y = 2
5 * 2 = 10
x = 10, y = 1
10 * 1 = 10
x = 15, y = 2/3
15 * (2/3) = 10
x = 20, y = 1/2
20 * (1/2) = 10
The products are all equal to 10, which means the relation satisfies inverse variation.
To represent the situation as an equation for inverse variation, we can use the formula y = k/x, where k is the constant of variation. Plugging in the given values, we can solve for k:
2 = k/5
k = 10
Now we can write the equation for this relation:
y = 10/x
In conclusion, the relation in the table represents inverse variation, and the equation that represents the situation is y = 10/x.