Explainer: Direct Variation

In this explainer, we will learn how to describe direct variation between two variables and use this to solve word problems.

β€œDirect variation” between two variables means that the two variables are directly proportional. If the two variables are the input and output of a function, then this function is a particular type of linear function.

A linear function is defined as a function that is represented by a line. If a function is represented by a line, this means that the rate of change (i.e., the ratio of the change in 𝑦 to the change in π‘₯ between any two points) is constant. The equation of a linear function is in the form 𝑦=π‘šπ‘₯+𝑏, where π‘š is the rate of change of the function and the slope of the line representing this function and 𝑏 is the 𝑦-intercept of the line.

Let us have a look at the figure displaying the two lines 𝑦=2π‘₯+3 and 𝑦=2π‘₯ and the corresponding function tables.

π‘₯0123
𝑦=2π‘₯0246
𝑦=2π‘₯+33579

The slope of both lines is the same: the lines are indeed parallel, and every time π‘₯ increases by 1, 𝑦 increases by 2 for both lines. However, there is a striking difference: for 𝑦=2π‘₯, we notice that 𝑦 is proportional to π‘₯, with a constant of proportionality of 2. This is actually what the equation tells us: 𝑦=2π‘₯. In other words, 𝑦 is the double of π‘₯.

We can generalize this observation and state that a line going through the origin represents a particular type of linear functions for which not only the rate of change but also simply the ratio of 𝑦 to π‘₯ is constant: it is equal to the slope of the line. The equation of these particular functions is in the form 𝑦=π‘šπ‘₯, showing the proportional relationship between 𝑦 and π‘₯, where π‘š is the coefficient of proportionality.

This type of relationship between two variables is also called a direct variation.

Definition: Direct Variation or Direct Proportion

Two variables are said to be in direct variation, or direct proportion, if their ratio is constant.

This type of relationship is often written as π‘¦βˆπ‘₯. It is mathematically described as 𝑦=π‘˜π‘₯, where π‘˜ is called the constant of variation or constant of proportionality.

By dividing both sides of the previous equation by π‘₯, we see that π‘˜ is the ratio of 𝑦 to π‘₯: π‘˜=𝑦π‘₯.

Let us look at a couple of examples where we are going to solve proportion equations.

Example 1: Solving Proportion Equations Involving Direct Variation

If π‘¦βˆπ‘₯ and π‘₯=75 when 𝑦=25, find the value of 𝑦 when π‘₯=30.

Answer

Recall that when π‘¦βˆπ‘₯ then 𝑦=π‘˜π‘₯, where π‘˜ is a constant.

So, π‘₯=75 when 𝑦=25: 25=π‘˜(75)π‘˜=13.𝑦=13π‘₯.

So, when π‘₯=30: 𝑦=13(30)=10.

Therefore, when π‘₯=30, 𝑦=10.

Example 2: Finding the Constant of Proportionality

If π‘¦βˆπ‘₯ and 𝑦=14 when π‘₯=6, determine the constant of proportionality.

Answer

Recall that π‘¦βˆπ‘₯ means that 𝑦=π‘˜π‘₯, where π‘˜ is the constant of proportionality. Since 𝑦=14 when π‘₯=6, we have 14=π‘˜β‹…6.

Dividing both sides by 6, we find π‘˜=146=73.

Hence, the constant of proportionality is 73.

In the next couple of examples, we are going to identify whether two variables are in direct variation or not.

Example 3: Identifying Direct Variation

Which of the following relations represents a direct variation between the two variables π‘₯ and 𝑦?

  1. π‘₯5=𝑦4
  2. π‘₯6=3𝑦
  3. π‘₯𝑦=6
  4. 𝑦=π‘₯+2

Answer

Recall that a direct variation is described mathematically as 𝑦=π‘˜π‘₯.

We need to identify which of the given relations can be rewritten as 𝑦=π‘˜π‘₯.

Relation A:

In relation A, π‘₯5=𝑦4.

By multiplying both sides by 4, we get 4β‹…π‘₯5=𝑦.

As multiplication is commutative, this can be rewritten as 45π‘₯=𝑦𝑦=45π‘₯.or

This relation represents a direct variation.

Relation B:

In relation B, π‘₯6=3𝑦.

By multiplying both sides by 𝑦, we get π‘₯𝑦6=3.

By dividing both sides by π‘₯, we find 𝑦6=3π‘₯.

And multiplying by 6 gives 𝑦=18π‘₯.

This does not describe a direct variation.

Relation C:

In relation C, π‘₯𝑦=6.

By dividing both sides by π‘₯, we find 𝑦=6π‘₯.

This does not describe a direct variation.

Relation D:

In relation D, 𝑦=π‘₯+2.

This is in the form 𝑦=π‘šπ‘₯+𝑏. This describes a linear function but not a direct variation since the constant 𝑏 is not zero. The corresponding line does not go through the origin.

The answer is, therefore, relation A.

Example 4: Identifying Direct Variation

Does 𝑦 vary directly with π‘₯? If so, what is the constant of variation?

π‘₯2468
𝑦3456

Answer

Recall that two variables are said to be in direct variation, or proportion, if their ratio is constant.

Let’s work out the ratio of 𝑦 to π‘₯ for all the pairs given.

π‘₯2468
𝑦3456
π‘₯𝑦2316543

The ratio of 𝑦 to π‘₯ is not constant. Therefore, 𝑦 does not vary directly with π‘₯.

Finally, let us look at a proportion equation in a real-world context.

Example 5: Solving Proportion Equations Involving Direct Variation

An object that weighs 120 N on Earth weighs 20 N on the Moon. Given that the weight of an object on Earth is directly proportional to its weight on the Moon, find the weight of the object on the Moon given that its weight on Earth is 126 N.

Answer

It is said here that the body’s weight on Earth is proportional to its weight on the Moon. It means that the ratio of the weight on the Moon to the weight on Earth is constant. This can be mathematically written as π‘Šπ‘Š=π‘˜,MoonEarth where π‘˜ is the constant of proportionality.

We can find the value of π‘˜ by using the first pair of values given in the question: π‘Š=20MoonN and π‘Š=120EarthN. Hence, using the equation given above and substituting this pair of values, we find π‘˜=20120=16.

Therefore, we have π‘Šπ‘Š=16MoonEarth, which can be also written as π‘Š=16π‘ŠMoonEarth.

We have found that the weight of the given body on the Moon is one sixth of its weight on Earth. Hence, the weight on the Moon of a body whose weight on Earth is 126 N is π‘Š=16Γ—126=21.MoonN

Key Points

  1. Two variables are said to be in direct variation, or direct proportion, if their ratio is constant.
  2. Direct variation between two variables 𝑦 and π‘₯ is often written as π‘¦βˆπ‘₯. It is mathematically described as 𝑦=π‘˜π‘₯, where π‘˜ is called the constant of variation or constant of proportionality.
  3. While linear relationships between two variables 𝑦 and π‘₯ are mathematically described as 𝑦=π‘šπ‘₯+𝑏, only those where 𝑏=0 correspond to direct variation (i.e., where 𝑦 and π‘₯ are proportional).
  4. The graph of direct variation (of a proportional relationship) is a straight line that goes through the origin.

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