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Direct Variation Formula

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When one quantity is directly reliant on the other, such as when one quantity grows in relation to the other and vice versa, the two variables are said to have a direct variation between them. It is a relationship between two variables in which one is a constant multiple of the other. Because the two variables are proportional to each other, they are sometimes referred to be directly proportional.

Example: The speed of a car and the distance it travels are an example of direct variation. When the speed is increased, the distance travelled in a given amount of time increases as well. Similarly, as the car’s speed lowers, the distance covered in that time interval reduces as well.

The direct variation formula connects two numbers by establishing a mathematical relationship in which one variable is a constant multiple of the other. It is given as follows:

y = kx

where x and y are the quantities in direct proportion to each other and k is a constant.

Solving a Direct Variation

The formula y = kx is used to solve a direct variation. If the proportionality constant needs to be determined, divide y by x to get the answer. If k is known and either x or y must be found, these values can be replaced into the equation above to discover the unknown value.

Example: Find constant of proportionality if x = 69 and y = 23 have a direct variation.

Solution: 

The formula for direct variation is y = kx or k = y/x.

Hence, k = 69/23 = 1/3.

Hence the constant of proportionality is 1/3.

Sample Problems

Problem 1. Suppose y varies directly as x, and y = 72 when x = 8. Write a direct variation equation that relates x and y.

Solution:

We know that the direct variation formula is y = kx.

Replace y with 72 and x with 8.

72 = k(8)

or, k = 72/8

k = 9.

Thus, the direct variation equation is y = 9x.

Problem 2. Using the equation obtained in the above problem, find x when y = 63.

Solution:

As per the problem above, the direct variation equation is y = 9x.

Replace y with 63.

63 = 9x

or, x = 63/9

x = 7.

Thus, x = 7, when y = 63.

Problem 3. The distance a jet travels vary directly as the number of hours it flies. Suppose it travelled 3420 miles in 6 hours. Write a direct variation equation for the distance d flown in time t.

Solution:

We know, Distance = Rate × Time

Let rate be r.

As per the given info, we have

⇒ 3420 = r × 6

⇒ r = 3420/6

⇒ r = 570

Thus, the direct variation equation is d = 570t.

Problem 4. In the above problem, estimate how many hours it will take for an airline to fly 6500 miles.

Solution:

As per the above problem, d = 570t.

Replace d with 6500.

6500 = 570t

t = 6500/570

t ≈ 11.4

It would take the airline approximately 11.4 hours to fly 6500 miles.

Problem 5. Suppose y varies directly as x, and y = 98 when x = 14. Write a direct variation equation that relates x and y.

Solution:

We know that the direct variation formula is y = kx.

Replace y with 98 and x with 14.

98 = k(14)

or, k = 98/14

k = 7.

Thus, the direct variation equation is y = 7x.

Problem 6. In the above problem, find y when x = -4.

Solution:

From the above problem, we have y = 7x.

Replace x with -4 in the equation.

y = 7(-4)

y = -28

Thus, y = -28 when x = -4.

Problem 7. If you post 5 messages on a message board, you receive 12 messages in return. Write a direct variation representing this info.

Solution:

We know that the direct variation formula is y = kx.

Replace y with 12 and x with 5.

12 = k(5)

or, k = 12/5

Thus, the direct variation equation is y = \frac{12}{5}x   .


Last Updated : 23 Jan, 2024
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