Direct Variation (also known as Direct Proportion)

The concept of direct variation is summarized by the equation below.

y=kx where k is the constant of variation

We say that y varies directly with x if y is expressed as the product of some constant number k and x.


Cases of Direct Variation

However, the value of k can’t equal zero, i.e. k \ne 0.

Case 1: k > 0 (k is positive)

If x increases then the value of y also increases, or if x decreases then the value of y also decreases.

Case 2: k < 0 (k is negative)

If x increases then the value of y decreases, or if x decreases then the value of y increases.

If we isolate k on one side, it reveals that k is the constant ratio between y and x. In other words, dividing y by x always yields a constant output.

k=y/x

k is also known as the constant of variation, or constant of proportionality.


Examples of Direct Variation

Example 1: Tell whether y varies directly with x in the table below. If yes, write an equation to represent the direct variation.

a table with two columns. the x-column has entries 3, 5, 7, and 9. the y-column has entries 6, 10, 14, and 18.

Solution:

To show that y varies directly with x, we need to verify if dividing y by x always gives us the same value.

a table with three columns. the x-column has entries 3, 5, 7 and 9. the y-column has entries 6, 10, 14, and 18. the k column has entries 2, 2, 2, and 2.

Since we always arrived at the same value of 2 when dividing y by x, we can claim that y varies directly with x. This constant number is, in fact, our k = 2.

To write the equation of direct variation, we replace the letter k by the number 2 in the equation y = kx.

y=2x

When an equation that represents direct variation is graphed in the Cartesian Plane, it is always a straight line passing through the origin.

Think of it as the Slope-Intercept Form of a line written as

y = mx + b where b = 0

Here is the graph of the equation we found above.

the line y=2x is graphed showing the points (3,6), (5,10), (7,14), and (9,18).

Example 2: Tell whether y varies directly with x in the table below. If yes, write an equation to represent direct variation.

a table with two columns. the x-column has entries -4, -3, -2, and -1. the y-column has entries 1, 0.75, 0.5, and 0.25

Solution:

Divide each value of y by the corresponding value of x.

a table with three columns. the x column has entries -4, -3, -2, and -1. the y column has entries 1, 0.75, 0.50, and 0.25. the k column has entries -0.25, -0.25, -0.25, and -0.25.

The quotient of y and x is always k = - \,0.25. That means y varies directly with x. Here is the equation that represents its direct variation.

y=-0.25x

Here is the graph. By having a negative value of k implies that the line has a negative slope. As you can see, the line is decreasing from left to right.

In addition, since k is negative we see that when x increases the value of y decreases.

the graph of the line y=-0.25x on a plane passing through four points

Example 3: Tell whether if y directly varies with x in the table. If yes, write the equation that shows direct variation.

a table with two columns. the x column has entries 1/3, 1/4, 1/5 and 1/6. the y column has entries -2, -3/2, -1/5, and -1.

Solution:

Find the ratio of y and x, and see if we can get a common answer which we will call constant k.

a table with three columns. the x-column with entries 1/3, 1/4, 1/5, and 1/6. the y column has entries -2, -3/2, -1/5, and -1. the k column has entries -6, -6, -1, and -6.

It looks like the k-value on the third row is different from the rest. In order for it to be a direct variation, they should all have the same k-value.

The table does not represent direct variation, therefore, we can’t write the equation for direct variation.


Example 4:  Given that y varies directly with x. If x = 12 then y = 8.

  • Write the equation of direct variation that relates x and y.
  • What is the value of y when x = - \,9?

a) Write the equation of direct variation that relates x and y.

Since y directly varies with x, I would immediately write down the formula so I can see what’s going on.

y is equal to k times x

We are given the information that when x = 12 then y = 8. Substitute the values of x and y in the formula and solve k.

k is equal to two times 2/3

Replace the “k” in the formula by the value solved above to get the direct variation equation that relates x and y.

y=2/3 of x

b) What is the value of y when x = - \,9?

To solve for y, substitute x = - \,9 in the equation found in part a).

y equals negative six

Example 5: If y varies directly with x, find the missing value of x in

(-3,27) and (x,-18)

Solution:

We will use the first point to find the constant of proportionality k and to set up the equation y = kx.

x equals -3, y equals 27

Substitute the values of x and y  to solve for k.

k equals negative nine

The equation of direct proportionality that relates x and y is…

y equals negative nine x

We can now solve for x in (x - \,18) by plugging in y = - \,18.

x equals two

Example 6: The circumference of a circle (C) varies directly with its diameter. If a circle with the diameter of 31.4 inches has a radius of 5 inches,

  • Write the equation of direct variation that relates the circumference and diameter of a circle.
  • What is the diameter of the circle with a radius of 7 inches?

a) Write the equation of direct variation that relates the circumference and diameter of a circle.

We don’t have to use the formula y = k\,x all the time.  But we can use it to come up with a similar set-up depending on what the problem is asking.

The problem tells us that the circumference of a circle varies directly with its diameter, we can write the following equation of direct proportionality instead.

C = kd

The diameter is not provided but the radius is. Since the radius is given as 5 inches, that means, we can find the diameter because it is equal to twice the length of the radius. This gives us 10 inches for the diameter.

k=3.14

The equation of direct proportionality that relates circumference and diameter is shown below. Notice, k is replaced by the numerical value 3.14.

C=3.14d

b) What is the diameter of a circle with a radius of 7 inches?

Since the equation requires diameter and not the radius, we need to convert first the value of radius to diameter. Remember that diameter is twice the measure of a radius, thus 7 inches of radius is equal to 14 inches in diameter.

Now, we substitute d = 14 into the formula to get the answer for circumference.

C=43.96

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Inverse Variation