Direct Variation - solving using the constant of variation (the "k" method)

Written by Deloris Luthin

Thursday, 15 May 2008 20:29

A direct variation is an equation where both variables either increase at the same time or decrease at the same time.  There are two methods for solving a direct variation problem.  They are what I call the "k" method which involves finding the constant of proportionality or the shortcut method which involves writing a proportion and solving it. So lets examine the "k" method first. First of all, when y varies directly as x, the basic equation is: y = k*x.  Look below and you will see an example of a direct variation and how to solve it. Given that a varies directly as b, and a = 75 when b = 40, find a when b = 12.  Try to solve it and then look down below to see how it is worked out. First we write the equation a = k*b, and then we substitute the first a value and the first b value and then solve for k.   So it looks like this: 75 = k *40 75/40 = k   (divide by 40) 15/8 = k   (reduce the fraction by dividing both numbers by 5) Then rewrite the equation with k = 15/8, so we get a = 15/8 * b Now substitute the second value for b which is 12 and then solve for a.  We get: a= 15/8 * 12 a= 45/2 ( divide 4 into 8 and 12 and then multiply straight across) Lets try one that is a little more difficult.   Suppose p is directly proportional to (r - 2), and p = 20 when r = 6, find p when r = 12.  Try it on your own and then check your answer below. First we write the equation:  p = k* (r - 2).   Then we substitute the first p value and the first r value to find k again.  So we get: 20 = k * (6 - 2) 20 = k * 4 5  = k Then we substitute the value of k back in the very first equation and the second r value and then solve for p.  So we get: p = 4 * (12 -2) p = 4 * 10 p = 40 Now lets try solving a word problem that is a direct variation.  Consider this problem: The speed of an object falling from rest in a vacuum is directly proporional to the time it has fallen.  After an object has fallen for 1.5 s, its speed is 14.7 m/s.  What is its speed after it has fallen 5 s? Try writing an equation and solve it.  Check your answer down below. First we will let s stand for the speed  and t for the time.  So we get the following equation: s = k * t Then we substitute the first value for speed in for s and the first value for time in for t and solve for k. So we get: 14.7 = k * 1.5 49/5 = k Then lets substitute the value of k back in the original equation along with the second value of time for t and then solve for the speed s.  we get: s = (49/5) * 5 s = 49 and then we label our answer   49 m/s