Variation problems involve formulas which show the relationship between two or more variables. Direct variation is represented as a linear equation with one variable found by multiplying the other by a constant. For example, the formula for circumference of a circle is C = 2πR. Notice that C is always found by multiplying R by the constant 2π. In terms of variation, we say that C varies directly with R or C is directly proportional to R. In general, if y varies directly with x, then y = kx for some constant k, (k ≠ 0). The constant k is known as the constant of variation.

There is a step by step process to solve problems involving variation. First translate the verbal model into an equation. Substitute the values given into the equation and solve for the constant k. Put the value of k back into the equation and all other remaining values. Then solve for the unknown variable.

**Example**: y varies directly with x. When y is 10, x is 4. What is y when x = 7 ?

**Step 1:** Write the equation.

Since y varies directly with x, y = kx.

**Step 2:** Substitute the values in for x and y and solve for k.

10 = k(4)

10 = 4k

5/2 = k

**Step 3:** Substitute k back into the original equation as well as 7 for x and solve for y.

y = (5/2)(7)

y = 35/2

**Example:** y varies directly with x. When y = 4, x = 11. What is x when y = 9 ?

The equation is y = kx, now substitute 4 for y and 11 for x into the equation.

4 = 11(k)

4 = 11k, solve for k to get

4/11 = k

Now substitute 4/11 in for k and 9 for y into the original equation and solve for x.

9 = (4/11)x

99/4 = x (divided 9 by 4/11 which is the same as multiplying 9 by 11/4)

**Example:** Suppose you are interested in converting US dollars to Philippine pesos. The number of pesos received is directly proportional to the number of dollars to be exchanged. If 21 US dollars equals 894 pesos, how many pesos will you receive for $1,400 dollars?

Let y = number of pesos and x = number of US dollars and k = constant of variation.

The equation for this problem is 894 = k(21)

894 = 21k

42.57 = k

To find out how many pesos you will receive for $1,400, substitute 42.57 for k and $1,400 for x.

y = 42.57(1,400)

y = 59,598

Note the the equation for direct variation, y = kx, is linear. The graph of this is a straight line through the origin with a slope of k.

This guide should help assist anyone having confusion about direct variation.