Direct Variation and Inverse Variation

Introduction to Direct Variation

Direct Variation is considered to be a mathematical relationship that exists between the two variables which can be expressed through an equation in which one of the variables are equal to a constant than the other. This is an equation that is supposed to function through expressing direct variation as such would be compared to inverse variation. “For example, if y varies directly as x, and y = 6 when x = 2, the constant of variation is k = = 3. Thus, the equation describing this direct variation is y = 3x.”

What Is a Direct Variation Formula?

Direct variation is supposed to be a type of proportionality where the quantity is considered to be directly varied with respect to the change in the other quantity. Therefore, this is considered to imply that there is an increase in the quantity where the other quantity is experienced through a proportionate increase and such would create a decrease when the other quantity decreases or reduces. Therefore, it can be stated that the direct variation is supposed to be a linear relationship where the graph would be consisting of a straight line. However, if there are two other quantities of indirect variation the other one would remain constant as it would be the multiple of the other. The formula of the direct variation is considered to be “y ∝ x.”

For instance, “The given quantities are y1 = 30, x1 = 6, x2 = 100, y2 = ? Using direct variation formula we have the following expression.

y1 / x1 = y2 / x2

30/6 = y2 / 1005 = y2 / 100

y2 = 500

Therefore, the value of y when x = 100 is 500.”

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Graph of Direct Variation

The graph regarding direct variation is considered to be resulting through a straight line and due to such, the direct variation is supposed to represent a linear equation as such would be in the form of two variables. In addition to this, the linear equation is supposed to be given through y = kx. This ratio of change is also supposed to be Δy/Δx and such is considered to be equal to k. Thus, the change is considered to represent the slope of the line as the direct variation graph is to be given through the formula.

Thus, it can be stated that when one of the quantities increases or it decreases with the increase or decrease in another quantity it is known as a direct variation.

Difference Between Direct Variation and Inverse Variation

Direct Variation	Inverse Variation
The direct variation is considered to be something that increases as well as decreases with the other quantity directly.	On the other hand, it can be understood that the quantity is supposed to increase with the decrease in another quantity and vice versa, and such is deemed to result in an inverse variation.
The ratio regarding direct variation is supposed to be through the two quantities and this will always remain constant as the constant is supposed to be regarding the proportionality.	Whereas, it can be understood that the product of the two quantities is supposed to be always constant which implies that the constant would be regarding proportionality.
The direct variation is supposed to be expressed as y ∝ x.	However, the product of the two quantities in the case of inverse variation is supposed to be expressed through y ∝ 1 / x.
The direct variation formula is supposed to be y = kx.	On the other hand, the inverse proportion formula is supposed to be given as y = k / x.
The graph regarding the direct variation formula is considered to be in a straight line.	On the other hand, the graph of an inverse variation is supposed to be through a rectangular hyperbola.
Therefore, it can be understood that proportionality is supposed to be based on the two quantities as such follows a direct relationship.	Whereas, the coordinate plane needs to be something that would be distinguished through a rectangular hyperbola and not a straight line as the ratio of the two quantities cannot be direct but can be constant.

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Important Notes on Direct Variation

The direct variation is supposed to be an equation that is a linear equation and it is considered to have two variables which are supposed to be having the constant proportionality. Along with such, it has a graph that is a coordinate plane which is a straight line. The ratio is of the two quantities and such are considered to be constant. It is different and distinct from inverse variation because it provides the relationship between the two mathematical quantities where they are increasing with respect to the other quantity where the quantities need to be understood as constant. So, if you are looking for quality linear algebra assignment help online services, our assignment experts are your best bet.

Examples of Direct Variation

Example 1:

“Plot the graph of the direct variation y = 5x

Solution:

x 0 1 -1

y 0 5 -5

Example 2:

“For example, a worker may be paid according to the number of hours he worked. The two quantities x (the number of hours worked) and y (the amount paid) are related in such a way that when x changes, y changes proportionately such that the ratio remains a constant.

We say that y varies directly with x. Let us represent the constant by k, i.e.

y/x = k or y = kx where k ≠ 0

If y varies directly as x, this relation is written as y ∝ x and read as y varies as x. The sign “ ∝ ” is read “varies as” and is called the sign of variation.”

Example 3:

“If y varies directly as x and given y = 9 when x = 5, find:

1. a) the equation connecting x and y
2. b) the value of y when x = 15
3. c) the value of x when y = 6

Solution:

1. a) y ∝x i.e. y = kx where k is a constant

Substitute x = 5 and y = 9 into the equation:

y = x

1. b) Substitute x = 15 into the equation

y = = 27

1. c) Substitute y = 6 into the equation