Direct variation

Introduction

Direct variation, also known as direct proportionality, describes a specific linear relationship between two variables. By definition, the statement “y varies directly as x” may be expressed mathematically as follows:

y=kx

where k is a non-zero constant:
k= y/x

Note that k goes by many names, including the proportionality constant, constant of variation, constant of proportionality and coefficient of proportionality. These names mean the same thing: the ratio or proportion y/x must remain constant. When x doubles, y must also double. When x triples, y must also triples. When x halves, y must also halve. And so on.

The relationship is often denoted, using the ∝ symbol, as:

y∝x

Officially, it reads:

y varies directly as x

But if you want to be a tad different and cool, you can say that:

y is (directly) proportional to x.

The above two statements imply that y is a function of x. x is the independent variable. y is the dependent one. When you know x, you’ll know y.

Did you know? If y is a function of x, then for each x value in the domain, there must be one and only one y=f(x). In other words, for each possible value of x, there must be one and only one value of f(x).

As I will discuss in Property #2, the relationship is symmetric. So you can also say any of the followings:

x and y (or y and x) vary directly.
x and y (or y and x) are in direct variation.
x and y (or y and x) are proportional.

Properties

[Graphical property]

y varies directly as x if and only if y is a linear function of x that passes through the origin with non-zero slope.

[Show graphs of y = mx in two cases: m>0 and m<0. Each graph should show all the four quadrants and be made explicit that the straight line passes through the origin.] Proof: When y is a linear function of x, that means y can be expressed in the following format: y = f(x) = mx + c where m and c are constant. We can deduce that m is the slope, and (0, c) is the Y-intercept. Moreover, since the line passes through the origin (0, 0), c must be zero. And since the line’s slope is non-zero, it follows that m≠0. We can therefore simplify the equation to: y = mx where m is a non-zero constant. Same thing as “y varies directly as x.” See? [Symmetric property] a∝b if and only if b∝a. Proof: If a∝b, we know that there exists a non-zero constant m where a=mb. Alternatively, we can say that there exists a non-zero constant (1/m) where b=(1/m)a. Same thing as b∝a. Did you know? For every non-zero number m, there exists a multiplicative inverse or reciprocal of m: (1/m), which is also non-zero. The graph of the reciprocal function: y=1/x is a rectangular hyperbola. [Transitive property] If a∝b and b∝c, then a∝c. Proof: If a=mb "and" b=nc where m and n are non-zero constants, then we can substitute the latter equation into the former. a=mb=m(nc) a=(mn)c Since (mn) is also a constant, we can say that a∝c. If a∝b and c∝d, then ac∝bd. Proof: If a=mb "and" c=nd where m and n are non-zero constants, then we can multiply these two equations together. ac=(mb)(nd) ac=(mn)cd Since (mn) is also a constant, we can say that ac∝bd. If a∝b "and both" a^n 〖 "and" b〗^n can be defined, then a^n∝b^n. Proof: If a=mb where m is a non-zero constant, then we can raise to both sides of the equation to the nth power (given that n is also a constant). Of course, if a^n "or" b^n is undefined, then you are out of luck. a^n=〖(mb)〗^n a^n=〖(m〗^n)b^n Since 〖(m〗^n) is also a constant, we can say that a^n∝b^n. If a∝b "and" a∝c, then a∝kb+lc where k "and" l are non-zero constants. Proof: If a=mb "and" a=nc "where" m "and" n are non-zero constants, then: b=a/m c=a/n Substitute these into kb+lc: kb+lc=k(a/m)+ l(a/n) kb+lc=(k/m+l/n)a Since (k/m+l/n)" is a constant," it follows that kb+lc∝a. Due to the symmetric property, we also know that a∝kb+lc. If a∝b,"then" y/x=dy/dx. This property will be discussed in detail in Question 3. Exercise Question 1 The commission income that a devil salesman earns varies directly as his sales volume. Last month he earned £700 from selling 3,500 souls. Find the amount of commission income he has earned this month so far for having sold 2,000 souls. Solution: Direct method: Find the formula connecting commission (y) and number of souls (x), and use the formula to find y when x = 2,000. Let k denote the constant of variation. y=kx We know that y=700 when x=3,500: 700 = k (3,500) k=1/5 Now, assign the value of k back into the original equation: y=x/5 So when x=2,000… y=2,000/5 y = £400 Shortcut: Note that the question does not ask you to find the formula connecting distance and speed. So you can skip that part, and jump right into finding the answer. As I said earlier, direct variation means that the ratio or proportion y/x remains constant. y_1/x_1 =y_2/x_2 700/3,500=y_2/2,000 1/5=y_2/2,000 2,000/5=y_2 y_2=£400 Tip: Suppose 〖(x〗_1,y_1) "and" 〖(x〗_2,y_2) "are possible solutions for" y=kx. k=y/x=y_1/x_1 =y_2/x_2 y_1/x_1 =y_2/x_2 If we know "the values of" x_1,y_1 "and" 〖 y〗_2, then we can find the value of x_2 using the “rule of three.” x_2=(x_1 y_2)/y_1 Question 2 The number of John’s sexual encounters in any given week varies directly as the square of the number of trips he makes to a plastic surgery clinic. There was a week he went to the clinic twice and got laid 19 times. Find the number of John’s sexual encounters in a week he visits the clinic 10 times. Solution: Direct method: Find the formula connecting sexual encounters (y) and clinic visits (x), and use the formula to find y when x = 10. Let k denote the constant of variation. y=kx^2 We know that y=19 when x=2: 19 = k 〖(2)〗^2 k=19/4 Now, assign the value of k back into the original equation: y= 19/4 x^2 So when x=10… y=19/4 〖(10)〗^2 y = 475 times Shortcut: Since y varies directly as x2, the ratio or proportion y/x^2 must remain constant. y_1/(x_1^2 )=y_2/(x_2^2 ) 19/2^2 =y_2/〖10〗^2 19/2^2 〖10〗^2=y_2 y_2=19〖(10/2)〗^2= 475 Question 3 Prove that in a perfectly competitive market, average revenue almost always equals to marginal revenue. Notes for non-economists: in a perfectly competitive market, there are so many buyers and sellers. All firms are price takers (rather than price setters). We can assume that price is a constant. Solution: Let y be total revenue, x be quantity supplied. Because the price P is a constant, it is easy to see that total revenue varies directly as quantity sold. Total revenue = (price)(quantity sold) y = Px Average revenue is total revenue divided by the number of units sold. Average revenue = y/x = Px/x=P when x≠0 Marginal revenue is the additional revenue received from selling one more unit. Marginal revenue = dy/dx=(d(Px))/dx=P Therefore, average revenue = marginal revenue whenever sales take place (x>0).

Note: This is another property of direct variation.
The constant of variation k=y/x=dy/dx.