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Direct Variation

Direct variation is a type of proportionality wherein one quantity directly varies with respect to a change in another quantity. This implies that if there is an increase in one quantity then the other quantity will experience a proportionate increase. Similarly, if one quantity decreases then the other quantity also decreases. Direct variation is a linear relationship hence, the graph will be a straight line.

Further, if two quantities are in direct variation then one will be a constant multiple of the other. In this article, we will elaborate on direct variation, its definition, formula, graph and associated examples.

1. What is Direct Variation?
2. What Is Direct Variation Formula?
3. Direct Variation Graph
4. Difference Between Direct Variation and Inverse Variation
5. FAQs on Direct Variation

What is Direct Variation?

Direct variation exists between any two variables when one quantity is directly dependent on the other i.e. if one quantity increases with respect to the other quantity and vice versa. It is the relationship between two variables where one of the variables is a constant multiple of the other. Since the two variables are directly related to each other it is also termed as directly proportional.

Direct variation and inverse variation are two types of proportionalities. Proportionality refers to a relationship where two quantities are multiplicatively connected by a constant. In a direct variation, the ratio of the two quantities remains the same whereas in an inverse variation the product of the two quantities remains constant. Here we shall check in detail the definition and examples of direct variation.

Direct Variation Definition

Two quantities as said to follow a direct variation if both increase or decrease by the same factor. Thus, an increase in one quantity leads to an increase in the other while a decrease in one quantity leads to a decrease in the other. In other words, if the ratio of the first quantity to the second quantity is a constant term, then the quantities are said to be directly proportional to each other. This constant value is known as the coefficient or constant of proportionality.

Direct Variation Example

The following two quick examples are helpful for an easy understanding of this concept of direct variation.

Example I: The formula for the circumference of a circle is given by C = 2πr or C = πd. Here, r is the radius and d is the diameter. This is an example of a direct variation. Thus, the circumference of a circle and its corresponding diameter are in direct variation with π being the constant of proportionality.

Example II: The quantity of Iron boxes made is directly proportional to the number of iron blocks. The number of iron blocks needed for 40 boxes is 160. How many iron blocks are needed for a box?

In the given problem, the number of iron blocks needed for 40 boxes is referred to as y = 160, and the number of boxes is referred to as x = 40. The number of iron blocks needed for a box is k. Here we use the direct variation formula of y = kx.

160 = k × 40
k = 160/40
k = 4

Thus 4 iron blocks are needed for a box.

What Is Direct Variation Formula?

Direct variation formula helps relate two quantities, having a mathematical relationship such that one of the variables is a constant multiple of the other. When two quantities are directly proportional to each other or are in direct variation they are represented using the symbol “\(\propto\)”. Suppose there are two quantities x and y that are in direct variation then they are expressed as follows:

y \(\propto\) x

When the proportionality sign is removed then the direct variation formula is given as follows.

Direct Variation Formula: y = kx

Here k is the constant of proportionality. If x is not equal to zero then the value of the constant of proportionality can be given as k = y/x. Thus, the ratio of these two variables is always a constant number. Another way of expressing the direct variation equation is x = y / k. This means that x is directly proportional to y with the constant of proportionality equalling 1 / k.

The formula for the direction variation for a set of two quantities that are linearly dependent is as follows.

Let us understand the formula of direct variation with the help of a simple example. Example: Let us assume that y varies directly with x, and y = 30 when x = 6. What is the value of y when x = 100?

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.

Direct Variation Graph

The graph of two quantities in a direct variation will result in a straight line. Thus, direct variation represents a linear equation in two variables. The linear equation is given by y = kx. The ratio of change \(\frac{\Delta y}{\Delta x}\) is also equal to k. This change represents the slope of the line. The direct variation graph is given as follows:

Difference Between Direct Variation and Inverse Variation

The difference between direct variation and inverse variation provides the relationship between two mathematical quantities. The two quantities are said to be in direct variation if one quantity is proportionally increasing with respect to another quantity, and the two quantities are said to be in inverse variation if one quantity is increasing and the other quantity is simultaneously decreasing. The relationship between the two quantities, y, and x is as follows.

Inverse and direct variation are both types of proportionalities. The difference between inverse and direct variation is given in the table below:

Direct Variation Inverse Variation
When one quantity increases (or decreases) with an increase (or decrease) in another quantity it is a direct variation. When one quantity increases with a decrease in another quantity or vice versa it results in an inverse variation.
The ratio of the two quantities will always be constant (constant of proportionality). The product of the two quantities will always be a constant (constant of proportionality).
It is expressed as y \(\propto\) x y \(\propto\) 1 / x
The direct variation formula is y = kx The inverse variation formula is given by y = k / x
The graph of a direct variation is a straight line. The graph of an inverse variation is a rectangular hyperbola.

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Important Notes on Direct Variation

  • A direct variation is a proportionality relationship in which two quantities follow a direct relationship. This implies that an increase (or decrease) in one quantity leads to a corresponding increase (or decrease) in another quantity.
  • The direct variation equation is a linear equation in two variables and is given by y = kx where k is the constant proportionality.
  • The direct variation graph in a coordinate plane is a straight line.
  • The ratio of two quantities in a direct variation is a constant.

Examples on Direct Variation

  1. Example 1: Plot the graph of the direct variation y = 5x
    Solution:

    x 0 1 -1 y 0 5 -5
  2. Example 2: If x = 10 and y = 20 follow a direct variation then find the constant of proportionality
    Solution: As x and y are in a direct variation thus y = kx or k = y / x.
    k = 20 / 10 = 2
    Answer: k = 2
  3. Example 3: Let x and y be in direct variation, x = 6 and y = 21. Then find the direct variation equation.
    Solution: As x and y are in a direct variation thus y = kx or k = y / x.
    k = 21 / 6 = 7 / 2
    y = \(\frac{7}{2}x\)
    Answer: The direct variation equation is y = \(\frac{7}{2}x\)

Practice Questions on Direct Variation

FAQs on Direct Variation

What is Direct Variation?

When one quantity directly varies with respect to another quantity it is known as direct variation. This implies that if one quantity increases or decreases the other quantity also increases or decreases proportionately.

What is the Direct Variation Equation?

The direct variation equation is given as y = kx where y and x are the two varying quantities and k is the constant of proportionality.

What is a Real-Life Direct Variation Example?

One example of direct variation is the speed of a car and the distance covered by it. If the speed increases the distance traveled within a certain time will also increase. Similarly, if the speed of the car decreases the distance covered within that interval of time will also decrease.

How to Solve Direct Variation?

To solve questions on direct variation the formula used is y = kx. If the constant of proportionality needs to be determined then y is divided by x to get the result. Suppose k is given and either x or y need to be determined then these values can be substituted in the aforementioned equation to find the unknown value.

How to Tell if an Equation is a Direct Variation?

If an equation is not in the form of y = kx then it does not follow a direct variation. For example y = 3x + 5 is not of the form y = kx hence, it doesn’t represent a direct variation. On the other hand, y = 1.2x is of the form y = kx and denotes a direct variation.

How to Graph a Direct Variation Equation?

To graph a direct variation equation, y = kx, the steps are as follows:

  • Substitute x with numerical values.
  • Find the corresponding values of y using the direct variation formula.
  • Using these test points, plot a graph. The graph will be a straight line passing through the origin.

Is y = 2x a Direct Variation?

Yes. As y = 2x is of the form y = kx thus, it represents a direct variation. The value of y will increase or decrease with an increase or decrease in the value of x.Yes.

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