Algebra 2

Inverse, joint and combined variation

When one quantity shrinks as another grows, or depends on several others at once, how do you write that as a single equation with one constant in it?

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Simplifying a rational expression and where it is undefinedAlgebra 2 Direct variationMath 8

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Inverse, joint and combined variation

Why this exists

Direct variation describes two quantities that grow together at a fixed rate: y = kx, so doubling x doubles y. Many of the laws people actually meet run the other way, or depend on more than one thing at once.

A fixed amount of work done faster takes less time. A gear with more teeth turns more slowly than the small gear driving it. Light from a lamp fades as you walk away from it, and it fades faster than the distance grows. The cost of painting a wall depends on its length and on its height, together. The pressure in a sealed tank rises when it is heated and falls when the tank is made bigger.

Each of these is one quantity that is a constant times a product or quotient of others. Once the pattern has a name and an equation, one measured case is enough to find the constant, and the constant then answers every other case. That is the whole use of variation: a single number carries the law.

Definition

Inverse variation. y varies inversely with x when, for some fixed number k \neq 0,

y = \frac{k}{x}, \qquad x \neq 0.

The number k is the constant of variation. Multiplying both sides by x gives the same statement as a product:

xy = k.

So y varies inversely with x exactly when the product of every pair of matching values is the same number. The word inverse means the reciprocal: \displaystyle y = k \cdot \frac{1}{x}, so y varies directly with \displaystyle \frac{1}{x}, which is why it is also written \displaystyle y \propto \frac{1}{x}. The same idea, met first through tables and stories, is inversely proportional relationships.

Inverse variation with a power. y varies inversely with the nth power of x when \displaystyle y = \frac{k}{x^n}, so that x^n y = k. “Inversely with the square” means \displaystyle y = \frac{k}{x^2}; “inversely with the square root” means \displaystyle y = \frac{k}{\sqrt{x}}.

Joint variation. z varies jointly with x and y when z = kxy for a fixed k \neq 0: z varies directly with each of them, and with their product. “Jointly” always means multiplied, never added.

Combined variation. Direct and inverse parts in one law, such as “z varies directly with x and inversely with y”:

z = \frac{kx}{y}, \qquad y \neq 0.

In every case the variables that appear in the numerator are the ones it varies directly with, and those in the denominator are the ones it varies inversely with. Since a denominator may not be zero, those variables carry the restriction from the domain of a rational expression.

Intuition

Take the lead example, \displaystyle y = \frac{84}{x}. A table makes the pattern plain.

x	1	2	3	4	6	7	12
y	84	42	28	21	14	12	7
xy	84	84	84	84	84	84	84

Going from x = 2 to x = 4 doubles x, and y falls from 42 to 21: it halves. Going from x = 4 to x = 12 triples x, and y falls from 21 to 7: it is divided by 3. Whatever factor multiplies x, the reciprocal of that factor multiplies y, so the product never moves.

The product has a picture. A point (x, y) in the first quadrant is the far corner of a rectangle with one corner at the origin, of width x and height y, and its area is xy. On the graph of \displaystyle y = \frac{84}{x} every such rectangle has area 84.

Figure 1Every point on \displaystyle y = \frac{84}{x} is the corner of a rectangle of area 84, however tall and thin or short and wide.

That picture also explains the shape. As x grows, the rectangle must get shorter to keep its area, so y gets close to 0 but never reaches it (area 84 needs some height). As x shrinks toward 0, the rectangle must get taller without limit. The curve hugs both axes and touches neither. It is the reciprocal function \displaystyle y = \frac{1}{x} stretched by the factor 84. Its points do not lie on a line: from (1, 84) to (2, 42) the slope is -42, but from (2, 42) to (4, 21) it is \displaystyle -\frac{21}{2}.

For joint variation, think of a wall: the paint needed varies jointly with length and height, because it is proportional to the area lh. Double the length and the paint doubles; double the height as well and it doubles again, four times in all.

Results

> Product test. y varies inversely with x if and only if xy is the same nonzero number > k for every pair of values. The constant is that product.

> Two pairs. If y varies inversely with x and (x_1, y_1), (x_2, y_2) are two pairs of > values, then x_1 y_1 = x_2 y_2.

> Scaling. If \displaystyle y = \frac{k}{x^n} and x is multiplied by c \neq 0, then y is multiplied by > \displaystyle \frac{1}{c^n}. With n = 1: double x, halve y. With n = 2: double x, divide y by 4.

> Joint and combined scaling. In \displaystyle z = \frac{k x^a y^b}{w^c}, multiplying x, y, w by > nonzero numbers p, q, r multiplies z by \displaystyle \frac{p^a q^b}{r^c}, whatever k is.

> Graph. For k > 0 the graph of \displaystyle y = \frac{k}{x} lies in the first and third quadrants; > for k < 0, in the second and fourth. It never meets either axis and is not a line.

Why it is true

Product test. If \displaystyle y = \frac{k}{x} with x \neq 0, multiplying both sides by x gives xy = k. Conversely, if xy = k for every pair and no x is 0, dividing by x gives \displaystyle y = \frac{k}{x}. The two statements say the same thing.

Two pairs. Each pair satisfies xy = k, so x_1 y_1 = k and x_2 y_2 = k, and two things equal to the same number are equal to each other.

Scaling. Replace x by cx:

\frac{k}{(cx)^n} = \frac{k}{c^n x^n} = \frac{1}{c^n} \cdot \frac{k}{x^n}.

So the new y is the old y times \displaystyle \frac{1}{c^n}. The joint and combined rule is the same calculation with the exponent laws applied to each variable in turn:

\frac{k (px)^a (qy)^b}{(rw)^c} = \frac{p^a q^b}{r^c} \cdot \frac{k x^a y^b}{w^c}.

Graph. The value x = 0 is excluded, so the graph never meets the y-axis. It meets the x-axis only where y = 0, which would need xy = 0 = k, and k \neq 0. When k > 0 the product xy is positive, so x and y have the same sign: both positive (first quadrant) or both negative (third). When k < 0 they have opposite signs. Finally, a line has the same slope between any two of its points, and the slopes -42 and \displaystyle -\frac{21}{2} found in the Intuition are different, so \displaystyle y = \frac{84}{x} is not a line; the same comparison works for any k.

Worked examples

Example 1

Finding the constant from one pair

y varies inversely with x, and y = 12 when x = 7. Write the equation, then find y when x = 16 and find x when y = 3.

Solution

Inverse variation means xy = k, so the one known pair gives the constant:

k = xy = 7 \cdot 12 = 84.

The equation is \displaystyle y = \frac{84}{x}. Now each question is a substitution. When x = 16,

y = \frac{84}{16} = \frac{21}{4}.

When y = 3, the product form is quicker: 3x = 84, so x = 28. Check: 28 \cdot 3 = 84, and \displaystyle 16 \cdot \frac{21}{4} = 84, so both new pairs have the right product.

Example 2

Is it inverse variation?

Decide whether each table shows y varying inversely with x. If it does, write the equation.

x	2	3	5	9
y (table A)	45	30	18	10
y (table B)	16	14	10	2

Solution

The product test is the whole question. For table A the products are 2 \cdot 45 = 90, 3 \cdot 30 = 90, 5 \cdot 18 = 90 and 9 \cdot 10 = 90: all the same, so y varies inversely with x, and \displaystyle y = \frac{90}{x}.

For table B the products are 32, 42, 50 and 18. They are not all equal, so this is not inverse variation, even though y falls as x rises. In fact the table follows the line y = 20 - 2x: each step of 1 in x lowers y by 2 (20 - 4 = 16, 20 - 6 = 14, 20 - 10 = 10, 20 - 18 = 2). Falling is not the test; a constant product is.

Example 3

Light from a lamp: an inverse square

The illuminance I (brightness on a surface, in lux) from a small lamp varies inversely with the square of the distance d in meters. At 2 meters it is 450 lux. Find the illuminance at 3 meters, and find the distance at which it is 50 lux.

Solution

“Inversely with the square” means \displaystyle I = \frac{k}{d^2}, so k = d^2 I:

k = 2^2 \cdot 450 = 1800, \qquad I = \frac{1800}{d^2}.

At d = 3, \displaystyle I = \frac{1800}{9} = 200 lux.

For I = 50, solve \displaystyle 50 = \frac{1800}{d^2}. Multiply both sides by d^2 (allowed, since d \neq 0) and divide by 50:

d^2 = \frac{1800}{50} = 36.

So d = 6 or d = -6, and a distance cannot be negative, so d = 6 meters. The scaling rule agrees: from 2 to 6 meters the distance is tripled, so the light is divided by 3^2 = 9, and \displaystyle \frac{450}{9} = 50.

Figure 2The lamp in Example 3. Tripling the distance from 2 to 6 meters divides the light by 9, not by 3.

Example 4

Two meshing gears

Two gears mesh. The speed of a gear, in turns per minute, varies inversely with its number of teeth. A gear with 24 teeth turns at 150 turns per minute and drives a gear with 40 teeth. How fast does the larger gear turn? How many teeth would a gear need to turn at 200 turns per minute?

Solution

The reason for the law is worth seeing. Where the gears touch, every tooth of one passes a tooth of the other, so in one minute the same number of teeth go past on both. For a gear that number is (teeth) times (turns), so the product n r is the same for every gear in the pair. Here it is 24 \cdot 150 = 3600, so \displaystyle r = \frac{3600}{n}.

Using two pairs, 24 \cdot 150 = 40 \cdot r, so

r = \frac{3600}{40} = 90 \text{ turns per minute}.

For 200 turns per minute, 200n = 3600, so n = 18 teeth. The fewer the teeth, the faster the gear, as the inverse law says.

Example 5

Painting a wall: joint variation

The cost C of painting a rectangular wall varies jointly with its length l and height h, in meters. A wall 4 meters long and 2.5 meters high costs \$150. What does a wall 6 meters long and 3 meters high cost, and what does the constant mean?

Solution

Joint variation means C = klh, so \displaystyle k = \frac{C}{lh}:

k = \frac{150}{4 \cdot 2.5} = \frac{150}{10} = 15, \qquad C = 15lh.

For the second wall, C = 15 \cdot 6 \cdot 3 = 270, so it costs \$270.

Because lh is the area of the wall, k = 15 is the price per square meter: \$15. The scaling rule gives a check: the length is multiplied by \displaystyle \frac{6}{4} = \frac{3}{2} and the height by \displaystyle \frac{3}{2.5} = \frac{6}{5}, so the cost is multiplied by \displaystyle \frac{3}{2} \cdot \frac{6}{5} = \frac{9}{5}, and \displaystyle 150 \cdot \frac{9}{5} = 270.

Example 6

Gas in a tank: combined variation

The pressure P of a gas, in kilopascals, varies directly with its temperature T in kelvin and inversely with its volume V in liters. At 300 kelvin in 6 liters the pressure is 200 kilopascals. Find the pressure at 360 kelvin in 8 liters.

Solution

T goes in the numerator and V in the denominator: \displaystyle P = \frac{kT}{V}. Multiply both sides by V and divide by T to isolate the constant:

k = \frac{PV}{T} = \frac{200 \cdot 6}{300} = 4, \qquad P = \frac{4T}{V}.

Then at T = 360 and V = 8,

P = \frac{4 \cdot 360}{8} = 180 \text{ kilopascals}.

The heating alone would raise the pressure by the factor \displaystyle \frac{360}{300} = \frac{6}{5}, and the larger tank alone would lower it by \displaystyle \frac{6}{8} = \frac{3}{4}. Together \displaystyle \frac{6}{5} \cdot \frac{3}{4} = \frac{9}{10}, and \displaystyle 200 \cdot \frac{9}{10} = 180: the bigger tank wins.

Example 7

Changing three things at once, without the constant

The force of gravity between two bodies varies jointly with their masses m_1 and m_2 and inversely with the square of the distance d between them:

F = \frac{G m_1 m_2}{d^2}.

If one mass is doubled, the other tripled, and the distance doubled, what happens to the force? And if both masses are doubled, how must the distance change for the force to stay the same?

Solution

No value of G is needed, because the scaling rule works for every constant. Replace m_1 by 2m_1, m_2 by 3m_2 and d by 2d:

\frac{G (2m_1)(3m_2)}{(2d)^2} = \frac{6}{4} \cdot \frac{G m_1 m_2}{d^2} = \frac{3}{2} F.

The force is multiplied by \displaystyle \frac{3}{2}.

Doubling both masses multiplies the numerator by 4. To keep F the same, the denominator d^2 must also be multiplied by 4, so d is multiplied by 2 (the positive square root, since a distance is positive). The bodies must be twice as far apart.

Boundaries

The constant is not zero. With k = 0 the “law” says y = 0 for every x, which is not a relationship between the two at all, so the definition requires k \neq 0.
No variable in a denominator may be zero. \displaystyle y = \frac{k}{x} says nothing at x = 0, and \displaystyle z = \frac{kx}{y} says nothing at y = 0. No value is assigned there, and the graph of \displaystyle y = \frac{k}{x} breaks into two pieces, with the y-axis as a vertical asymptote.
Falling is not enough. A quantity that decreases as another increases need not vary inversely with it; y = 20 - 2x in Example 2 falls but has no constant product. The test is xy = k, every time.
A shifted reciprocal is not inverse variation. \displaystyle y = \frac{84}{x} + 5 and \displaystyle y = \frac{84}{x - 2} have the same shape of graph, but their products xy are not constant. Those belong to transformed reciprocal graphs.
“Double, halve” is the first power only. For \displaystyle y = \frac{k}{x^2}, doubling x divides y by 4, and for \displaystyle y = \frac{k}{\sqrt{x}} it divides y by \displaystyle \sqrt{2}.
Jointly means multiplied. z = kxy is joint variation; z = kx + ky is not, since doubling both x and y only doubles it.
A model holds where it was measured. A law fitted to real data is only claimed over the range of values it was checked on. The lamp formula gives an unlimited brightness as d approaches 0, which no real lamp produces, because up close the lamp is no longer a point.

Where this goes next

The graph of \displaystyle y = \frac{k}{x} is the reciprocal function stretched by k; shifting it left, right, up or down gives the transformed reciprocal graphs, whose asymptotes move with it. Finding an unknown in a variation law, as in Example 3, is a small case of solving a rational equation: multiply through by the denominator, then solve what is left. Joint and combined variation are how most formulas in physics and geometry are first read: the area of a triangle varies jointly with base and height, and the period of a pendulum varies directly with the square root of its length. Reading which variables sit above the line and which below tells you, before any number is used, what each change does.

Exercises

Recognizing inverse variation

Does this table show y varying inversely with x? If so, give the equation. x: 2, 4, 7, 8; y: 28, 14, 8, 7.
Does this table show y varying inversely with x? x: 2, 4, 5, 10; y: 25, 15, 12, 6.
Show that xy = -32 is an inverse variation, and give its constant.
Is \displaystyle y = \frac{7}{x} + 2 an inverse variation? Use the product test with x = 1 and x = 7.

Equations and missing values

y varies inversely with x, and y = 9 when x = 7. Write the equation and find y when x = 21.
y varies inversely with x, and y = -5 when x = 7. Find x when y = 2.
y varies inversely with x, and \displaystyle y = \frac{2}{3} when x = 21. Find y when x = 8.
y varies inversely with the square of x, and y = 10 when x = 3. Find y when x = 6.
y varies inversely with the square root of x, and y = 6 when x = 16. Find y when x = 9.
y varies inversely with the cube of x. If x is tripled, what happens to y?

Joint and combined variation

z varies jointly with x and y, and z = 195 when x = 3 and y = 5. Find z when x = 4 and y = 2.
w varies jointly with x and the square of y, and w = 135 when x = 5 and y = 3. Find w when x = 2 and y = 5.
y varies directly with x and inversely with z, and y = 10 when x = 4 and z = 6. Find y when x = 8 and z = 20.
y varies directly with the square of x and inversely with z, and y = 12 when x = 2 and z = 5. Find y when x = 3 and z = 9.
Write an equation for “a varies jointly with b and c and inversely with the square of d”, and say which variable may not be zero.

Stories

The pitch (frequency) of a plucked string varies inversely with its length. A string 60 centimeters long sounds at 220 hertz. What is the frequency when it is shortened to 44 centimeters, and what length gives 330 hertz?
The strength of a radio signal varies inversely with the square of the distance from the tower. At 5 kilometers the strength is 0.36 units. Find the strength at 15 kilometers, and the distance at which it is 1 unit.
Simple interest varies jointly with the amount invested P and the number of years t. \$2000 invested for 3 years earns \$240. What does \$3500 earn in 5 years, and what does the constant mean?
The load a wooden beam can safely carry varies jointly with its width and the square of its depth, and inversely with its length. A beam 2 inches wide, 4 inches deep and 10 feet long carries 640 pounds. What does a beam 3 inches wide, 6 inches deep and 15 feet long carry?
\displaystyle y = \frac{kx}{z^2}. If x is doubled and z is halved, by what factor is y multiplied?

Reasoning

Explain why the graph of \displaystyle y = \frac{k}{x} with k > 0 never crosses either axis.
Give a relationship in which y decreases whenever x increases but y does not vary inversely with x, and show that it fails the product test.
Show that if y varies inversely with x, and x varies inversely with z, then y varies directly with z.
Show that if y varies inversely with x, then \displaystyle \frac{1}{y} varies directly with x, and say what the graph of \displaystyle \frac{1}{y} against x looks like.
z varies jointly with x and y. Show that doubling x and halving y leaves z unchanged.

Answers

Yes, \displaystyle y = \frac{56}{x}. The products are 2 \cdot 28 = 56, 4 \cdot 14 = 56, 7 \cdot 8 = 56 and 8 \cdot 7 = 56, all equal.
No. The products are 50, 60, 60, 60; the first differs, so there is no single constant.
Dividing by x \neq 0 gives \displaystyle y = -\frac{32}{x}, of the form \displaystyle \frac{k}{x} with k = -32.
No. xy = 7 + 2x, which is 9 at x = 1 but 21 at x = 7; the product is not constant.
k = 7 \cdot 9 = 63, so \displaystyle y = \frac{63}{x}, and at x = 21, \displaystyle y = \frac{63}{21} = 3.
k = 7 \cdot (-5) = -35, so 2x = -35 and \displaystyle x = -\frac{35}{2}.
\displaystyle k = 21 \cdot \frac{2}{3} = 14, so \displaystyle y = \frac{14}{8} = \frac{7}{4}.
k = 3^2 \cdot 10 = 90, so \displaystyle y = \frac{90}{6^2} = \frac{90}{36} = \frac{5}{2}.
\displaystyle k = \sqrt{16} \cdot 6 = 24, so \displaystyle y = \frac{24}{\sqrt{9}} = \frac{24}{3} = 8.
y is multiplied by \displaystyle \frac{1}{27}: \displaystyle \frac{k}{(3x)^3} = \frac{1}{27} \cdot \frac{k}{x^3}.
\displaystyle k = \frac{195}{3 \cdot 5} = 13, so z = 13xy and z = 13 \cdot 4 \cdot 2 = 104.
w = kxy^2 with \displaystyle k = \frac{135}{5 \cdot 9} = 3, so w = 3 \cdot 2 \cdot 25 = 150.
\displaystyle y = \frac{kx}{z} with \displaystyle k = \frac{10 \cdot 6}{4} = 15, so \displaystyle y = \frac{15 \cdot 8}{20} = 6.
\displaystyle y = \frac{kx^2}{z} with \displaystyle k = \frac{12 \cdot 5}{4} = 15, so \displaystyle y = \frac{15 \cdot 9}{9} = 15.
\displaystyle a = \frac{kbc}{d^2} with k \neq 0. b and c are in the numerator (direct), d^2 in the denominator (inverse), so d may not be zero.
\displaystyle f = \frac{k}{L} with k = 60 \cdot 220 = 13200. At 44 centimeters, \displaystyle f = \frac{13200}{44} = 300 hertz. For 330 hertz, \displaystyle L = \frac{13200}{330} = 40 centimeters.
\displaystyle S = \frac{k}{d^2} with k = 5^2 \cdot 0.36 = 9. At 15 kilometers, \displaystyle S = \frac{9}{225} = 0.04 units. For S = 1, d^2 = 9 and d = 3 kilometers (the positive root).
I = kPt with \displaystyle k = \frac{240}{2000 \cdot 3} = 0.04, so I = 0.04 \cdot 3500 \cdot 5 = 700: \$700. The constant is the yearly interest rate, 4\% of the amount per year.
\displaystyle S = \frac{kwd^2}{L} with \displaystyle k = \frac{640 \cdot 10}{2 \cdot 16} = 200, so \displaystyle S = \frac{200 \cdot 3 \cdot 36}{15} = 1440 pounds.
y is multiplied by 8: \displaystyle \frac{k(2x)}{(z/2)^2} = \frac{2}{1/4} \cdot \frac{kx}{z^2} = 8 \cdot \frac{kx}{z^2}.
Crossing the y-axis would need a point with x = 0, where \displaystyle \frac{k}{x} is not defined. Crossing the x-axis would need y = 0, so xy = 0, but every point has xy = k > 0.
For example y = 10 - x: it falls as x rises, but the points (1, 9) and (2, 8) have products 9 and 16, which differ, so it is not inverse variation.
\displaystyle y = \frac{k}{x} and \displaystyle x = \frac{m}{z} with k, m \neq 0. Substituting, \displaystyle y = k \div \frac{m}{z} = \frac{k}{m} z, which is direct variation with constant \displaystyle \frac{k}{m} \neq 0.
From \displaystyle y = \frac{k}{x}, \displaystyle \frac{1}{y} = \frac{x}{k} = \frac{1}{k} x, which is direct variation with constant \displaystyle \frac{1}{k}. Its graph is a line through the origin with slope \displaystyle \frac{1}{k}, with the point at x = 0 missing, since y is not defined there.
\displaystyle k(2x)\left(\frac{y}{2}\right) = k \cdot 2 \cdot \frac{1}{2} \cdot xy = kxy, so z is unchanged: the factors 2 and \displaystyle \frac{1}{2} multiply to 1.