Direct variation is a type of relationship between two variables where one variable is a constant multiple of the other. In mathematical terms, this relationship can be represented by the equation y = kx, where y and x are the variables, and k is the constant of variation. When given an ordered pair, such as (2, 7), we can determine whether or not it represents a direct variation relation by analyzing the equation and proving its validity.
Analyzing the Direct Variation Equation: (2, 7)
In the case of the ordered pair (2, 7), we can set up the direct variation equation as y = kx. Plugging in the values from the ordered pair, we get 7 = 2k. Solving for k, we find that k = 7/2 = 3.5. This means that the constant of variation for this particular relationship is 3.5. By analyzing the direct variation equation, we can see that the ordered pair (2, 7) does indeed represent a direct variation relation.
To further solidify our conclusion, we can also check if the other values in the dataset follow the same direct variation pattern. By plugging in other ordered pairs into the direct variation equation y = 3.5x, we can verify if they satisfy the relation. If all the ordered pairs in the dataset follow the same pattern, then we can confidently say that the relationship between the variables is a direct variation. In the case of the ordered pair (2, 7), since it satisfies the direct variation equation y = 3.5x, we can conclude that it represents a direct variation relation.
Proving the Direct Variation Relation with Ordered Pair (2, 7)
By proving that the ordered pair (2, 7) satisfies the direct variation equation y = 3.5x, we can confidently say that it represents a direct variation relation. This means that as one variable increases, the other variable also increases by a constant multiple. The constant of variation, in this case, is 3.5, indicating the rate at which the variables are related to each other. By utilizing mathematical principles and analyzing the equation, we can determine the nature of the relationship between the variables in the ordered pair (2, 7) and establish that it follows a direct variation pattern.
In conclusion, by analyzing the direct variation equation y = 3.5x and proving its validity with the ordered pair (2, 7), we have successfully determined that the relationship between the variables in the dataset follows a direct variation pattern. This method of analyzing and proving direct variation relations with ordered pairs allows us to understand how variables are related to each other and the constant factor by which they vary. By applying mathematical principles and calculations, we can confidently identify direct variation relations in different datasets and analyze the patterns between variables.