Constant variation seems like a very ironic term when combined.

After all, “constant” means one thing is not changing at all, while “variation” means changesIn math, constant variation has a deeper meaning regardless of the ironic termIn this article, you will how to get a constant of variation in both direct and inverse relationships.

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What is Constant of Variation?

Variation happens when the value of any variables are changed. For example, if variable x = 10, adding 2 to its value will make it x = 12. The addition of 2 is considered as the variation, as it is the change that happens to x. If the variation doesn’t change over time (e.g., x = 14, 16, 18…), then it is considered as a constant.

There are times when a change in a variable (X) affects the value of another variable (Y). Depending on their relationship (direct or inverse), the value of y will either decrease or increase, too, if x is changed. The unchanged ratio of the change between the two variables is referred to as the constant of variation. In this example, the ratio is 2:4, which remains the same throughout the set.

Direct Variation Inverse Variation
X 3 5 7 9 3 5 7 9
Y 6 10 14 18 18 14 10 6


Direct Variation

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Direct variation is a relationship that happens when the change in the first variable also happens to the second variable. Thus, if x increases in value, y’s value will also increase. This also applies when there’s a decrease. The formula expresses the k constant or the amount of change in the equation:

Inverse Variation

Inverse variation, on the other hand, happens when the change in the first variable has the opposite effect in the second variable. Thus, if you add 2 in x, y will decrease by 2. This formula expresses the equation:

Finding The “K” In Direct Variation

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Sometimes, the K ratio is not apparent. You can solve K via the given formula below, where x and y are the first and second variables. Here’s a simple example of getting the constant of variation when the relationship of two variables are direct.

If y varies directly as x and y = 10 when x = 5, what is the constant of variation (k)?

  1. Substitute the values

Substitute the given values in the formula for direct variation, y = kx. Since both x and y are given, all you have to do is to plug its values to the equation.

 y = kx

10 = (k)(5)

  1. Solve for the K using y = kx

Once the values are substituted, you can solve for the value of k. In this example, we have divided 5 into both sides to isolate the constant. The final answer is k = 2.

10 = (k)(5)

10/5 = (k)/(5)

 2 = k

Getting the constant of variation is quite easy if you got the hang of it. At its simplest explanation, it’s all about substituting the right value to the right equation. But depending on what the problem requires, you might have to solve for the specific value of x and y. Expanding this example, find y when x = 8; and find x when y = 15. You can do this by plugging both the k and x in the equation y = kx.

Find y when x = 8

x = 8, y = ? 

y = kx

y = (2)(8)

y = 16

Find x when y = 15

 x = ?, y = 15

 y = kx

15 = 2x

15/2 = 2x/2

15/2 = x

Finding The “K” In Inverse Variation

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For finding the constant in an inverse variation, the formula to be used is y = k/x. When the problem asks for specifics (what is x when y is __), the same formula used in Direct Variation is applied. But instead of using y = kx, the inverse variation formula (y = k/x) will be used.

Sample problem: If y varies inversely as x and y = 10 when x = 5, what is the constant of variation (k)? Using the formula y = k/x, this is the solution:

y = k/x

10 = k/5

10 x 5 = k

50 = k

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