Chapter 2: Linear Equations

# Direct Variation Problems

There are many mathematical relations that occur in life. For instance, a flat commission salaried salesperson earns a percentage of their sales, where the more they sell equates to the wage they earn. An example of this would be an employee whose wage is 5% of the sales they make. This is a direct or a linear variation, which, in an equation, would look like:

$\text{Wage }(x)=5\%\text{ Commission }(k)\text{ of Sales Completed }(y)$

or

$x=ky$

A historical example of direct variation can be found in the changing measurement of pi, which has been symbolized using the Greek letter π since the mid 18th century. Variations of historical π calculations are Babylonian $\left(\dfrac{25}{8}\right),$ Egyptian $\left(\dfrac{16}{9}\right)^2,$ and Indian $\left(\dfrac{339}{108}\text{ and }10^{\frac{1}{2}}\right).$ In the 5th century, Chinese mathematician Zu Chongzhi calculated the value of π to seven decimal places (3.1415926), representing the most accurate value of π for over 1000 years.

Pi is found by taking any circle and dividing the circumference of the circle by the diameter, which will always give the same value: 3.14159265358979323846264338327950288419716… (42 decimal places). Using an infinite-series exact equation has allowed computers to calculate π to 1013 decimals.

$\begin{array}{c} \text{Circumference }(c)=\pi \text{ times the diameter }(d) \\ \\ \text{or} \\ \\ c=\pi d \end{array}$

All direct variation relationships are verbalized in written problems as a direct variation or as directly proportional and take the form of straight line relationships. Examples of direct variation or directly proportional equations are:

• $x=ky$
• $x$ varies directly as $y$
• $x$ varies as $y$
• $x$ varies directly proportional to $y$
• $x$ is proportional to $y$
• $x=ky^2$
• $x$ varies directly as the square of $y$
• $x$ varies as $y$ squared
• $x$ is proportional to the square of $y$
• $x=ky^3$
• $x$ varies directly as the cube of $y$
• $x$ varies as $y$ cubed
• $x$ is proportional to the cube of $y$
• $x=ky^{\frac{1}{2}}$
• $x$ varies directly as the square root of $y$
• $x$ varies as the root of $y$
• $x$ is proportional to the square root of $y$

Example 2.7.1

Find the variation equation described as follows:

The surface area of a square surface $(A)$ is directly proportional to the square of either side $(x).$

Solution:

$\begin{array}{c} \text{Area }(A) =\text{ constant }(k)\text{ times side}^2\text{ } (x^2) \\ \\ \text{or} \\ \\ A=kx^2 \end{array}$

Example 2.7.2

When looking at two buildings at the same time, the length of the buildings’ shadows $(s)$ varies directly as their height $(h).$ If a 5-story building has a 20 m long shadow, how many stories high would a building that has a 32 m long shadow be?

The equation that describes this variation is:

$h=kx$

Breaking the data up into the first and second parts gives:

$\begin{array}{ll} \begin{array}{rrl} \\ &&\textbf{1st Data} \\ s&=&20\text{ m} \\ h&=&5\text{ stories} \\ k&=&\text{find 1st} \\ \\ &&\text{Find }k\text{:} \\ h&=&kx \\ 5\text{ stories}&=&k\text{ (20 m)} \\ k&=&5\text{ stories/20 m}\\ k&=&0.25\text{ story/m} \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ s&=&\text{32 m} \\ h&=&\text{find 2nd} \\ k&=&0.25\text{ story/m} \\ \\ &&\text{Find }h\text{:} \\ h&=&kx \\ h&=&(0.25\text{ story/m})(32\text{ m}) \\ h&=&8\text{ stories} \end{array} \end{array}$

# Inverse Variation Problems

Inverse variation problems are reciprocal relationships. In these types of problems, the product of two or more variables is equal to a constant. An example of this comes from the relationship of the pressure $(P)$ and the volume $(V)$ of a gas, called Boyle’s Law (1662). This law is written as:

$\begin{array}{c} \text{Pressure }(P)\text{ times Volume }(V)=\text{ constant} \\ \\ \text{ or } \\ \\ PV=k \end{array}$

Written as an inverse variation problem, it can be said that the pressure of an ideal gas varies as the inverse of the volume or varies inversely as the volume. Expressed this way, the equation can be written as:

$P=\dfrac{k}{V}$

Another example is the historically famous inverse square laws. Examples of this are the force of gravity $(F_{\text{g}}),$ electrostatic force $(F_{\text{el}}),$ and the intensity of light $(I).$ In all of these measures of force and light intensity, as you move away from the source, the intensity or strength decreases as the square of the distance.

In equation form, these look like:

$F_{\text{g}}=\dfrac{k}{d^2}\hspace{0.25in} F_{\text{el}}=\dfrac{k}{d^2}\hspace{0.25in} I=\dfrac{k}{d^2}$

These equations would be verbalized as:

• The force of gravity $(F_{\text{g}})$ varies inversely as the square of the distance.
• Electrostatic force $(F_{\text{el}})$ varies inversely as the square of the distance.
• The intensity of a light source $(I)$ varies inversely as the square of the distance.

All inverse variation relationship are verbalized in written problems as inverse variations or as inversely proportional. Examples of inverse variation or inversely proportional equations are:

• $x=\dfrac{k}{y}$
• $x$ varies inversely as $y$
• $x$ varies as the inverse of $y$
• $x$ varies inversely proportional to $y$
• $x$ is inversely proportional to $y$
• $x=\dfrac{k}{y^2}$
• $x$ varies inversely as the square of $y$
• $x$ varies inversely as $y$ squared
• $x$ is inversely proportional to the square of $y$
• $x=\dfrac{k}{y^3}$
• $x$ varies inversely as the cube of $y$
• $x$ varies inversely as $y$ cubed
• $x$ is inversely proportional to the cube of $y$
• $x=\dfrac{k}{y^{\frac{1}{2}}}$
• $x$ varies inversely as the square root of $y$
• $x$ varies as the inverse root of $y$
• $x$ is inversely proportional to the square root of $y$

Example 2.7.3

Find the variation equation described as follows:

The force experienced by a magnetic field $(F_{\text{b}})$ is inversely proportional to the square of the distance from the source $(d_{\text{s}}).$

Solution:

$F_{\text{b}} = \dfrac{k}{{d_{\text{s}}}^2}$

Example 2.7.4

The time $(t)$ it takes to travel from North Vancouver to Hope varies inversely as the speed $(v)$ at which one travels. If it takes 1.5 hours to travel this distance at an average speed of 120 km/h, find the constant $k$ and the amount of time it would take to drive back if you were only able to travel at 60 km/h due to an engine problem.

The equation that describes this variation is:

$t=\dfrac{k}{v}$

Breaking the data up into the first and second parts gives:

$\begin{array}{ll} \begin{array}{rrl} &&\textbf{1st Data} \\ v&=&120\text{ km/h} \\ t&=&1.5\text{ h} \\ k&=&\text{find 1st} \\ \\ &&\text{Find }k\text{:} \\ k&=&tv \\ k&=&(1.5\text{ h})(120\text{ km/h}) \\ k&=&180\text{ km} \end{array} & \hspace{0.5in} \begin{array}{rrl} \\ \\ \\ &&\textbf{2nd Data} \\ v&=&60\text{ km/h} \\ t&=&\text{find 2nd} \\ k&=&180\text{ km} \\ \\ &&\text{Find }t\text{:} \\ t&=&\dfrac{k}{v} \\ \\ t&=&\dfrac{180\text{ km}}{60\text{ km/h}} \\ \\ t&=&3\text{ h} \end{array} \end{array}$

# Joint or Combined Variation Problems

In real life, variation problems are not restricted to single variables. Instead, functions are generally a combination of multiple factors. For instance, the physics equation quantifying the gravitational force of attraction between two bodies is:

$F_{\text{g}}=\dfrac{Gm_1m_2}{d^2}$

where:

• $F_{\text{g}}$ stands for the gravitational force of attraction
• $G$ is Newton’s constant, which would be represented by $k$ in a standard variation problem
• $m_1$ and $m_2$ are the masses of the two bodies
• $d^2$ is the distance between the centres of both bodies

To write this out as a variation problem, first state that the force of gravitational attraction $(F_{\text{g}})$ between two bodies is directly proportional to the product of the two masses $(m_1, m_2)$ and inversely proportional to the square of the distance $(d)$ separating the two masses. From this information, the necessary equation can be derived. All joint variation relationships are verbalized in written problems as a combination of direct and inverse variation relationships, and care must be taken to correctly identify which variables are related in what relationship.

Example 2.7.5

Find the variation equation described as follows:

The force of electrical attraction $(F_{\text{el}})$ between two statically charged bodies is directly proportional to the product of the charges on each of the two objects $(q_1, q_2)$ and inversely proportional to the square of the distance $(d)$ separating these two charged bodies.

Solution:

$F_{\text{el}}=\dfrac{kq_1q_2}{d^2}$

Solving these combined or joint variation problems is the same as solving simpler variation problems.

First, decide what equation the variation represents. Second, break up the data into the first data given—which is used to find $k$—and then the second data, which is used to solve the problem given. Consider the following joint variation problem.

Example 2.7.6

$y$ varies jointly with $m$ and $n$ and inversely with the square of $d$. If $y = 12$ when $m = 3$, $n = 8$, and $d = 2,$ find the constant $k$, then use $k$ to find $y$ when $m=-3$, $n = 18$, and $d = 3$.

The equation that describes this variation is:

$y=\dfrac{kmn}{d^2}$

Breaking the data up into the first and second parts gives:

$\begin{array}{ll} \begin{array}{rrl} \\ \\ \\ && \textbf{1st Data} \\ y&=&12 \\ m&=&3 \\ n&=&8 \\ d&=&2 \\ k&=&\text{find 1st} \\ \\ &&\text{Find }k\text{:} \\ y&=&\dfrac{kmn}{d^2} \\ \\ 12&=&\dfrac{k(3)(8)}{(2)^2} \\ \\ k&=&\dfrac{12(2)^2}{(3)(8)} \\ \\ k&=& 2 \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ y&=&\text{find 2nd} \\ m&=&-3 \\ n&=&18 \\ d&=&3 \\ k&=&2 \\ \\ &&\text{Find }y\text{:} \\ y&=&\dfrac{kmn}{d^2} \\ \\ y&=&\dfrac{(2)(-3)(18)}{(3)^2} \\ \\ y&=&12 \end{array} \end{array}$

# Questions

For questions 1 to 12, write the formula defining the variation, including the constant of variation $(k).$

1. $x$ varies directly as $y$
2. $x$ is jointly proportional to $y$ and $z$
3. $x$ varies inversely as $y$
4. $x$ varies directly as the square of $y$
5. $x$ varies jointly as $z$ and $y$
6. $x$ is inversely proportional to the cube of $y$
7. $x$ is jointly proportional with the square of $y$ and the square root of $z$
8. $x$ is inversely proportional to $y$ to the sixth power
9. $x$ is jointly proportional with the cube of $y$ and inversely to the square root of $z$
10. $x$ is inversely proportional with the square of $y$ and the square root of $z$
11. $x$ varies jointly as $z$ and $y$ and is inversely proportional to the cube of $p$
12. $x$ is inversely proportional to the cube of $y$ and square of $z$

For questions 13 to 22, find the formula defining the variation and the constant of variation $(k).$

1. If $A$ varies directly as $B,$ find $k$ when $A=15$ and $B=5.$
2. If $P$ is jointly proportional to $Q$ and $R,$ find $k$ when $P=12, Q=8$ and $R=3.$
3. If $A$ varies inversely as $B,$ find $k$ when $A=7$ and $B=4.$
4. If $A$ varies directly as the square of $B,$ find $k$ when $A=6$ and $B=3.$
5. If $C$ varies jointly as $A$ and $B,$ find $k$ when $C=24, A=3,$ and $B=2.$
6. If $Y$ is inversely proportional to the cube of $X,$ find $k$ when $Y=54$ and $X=3.$
7. If $X$ is directly proportional to $Y,$ find $k$ when $X=12$ and $Y=8.$
8. If $A$ is jointly proportional with the square of $B$ and the square root of $C,$ find $k$ when $A=25, B=5$ and $C=9.$
9. If $y$ varies jointly with $m$ and the square of $n$ and inversely with $d,$ find $k$ when $y=10, m=4, n=5,$ and $d=6.$
10. If $P$ varies directly as $T$ and inversely as $V,$ find $k$ when $P=10, T=250,$ and $V=400.$

For questions 23 to 37, solve each variation word problem.

1. The electrical current $I$ (in amperes, A) varies directly as the voltage $(V)$ in a simple circuit. If the current is 5 A when the source voltage is 15 V, what is the current when the source voltage is 25 V?
2. The current $I$ in an electrical conductor varies inversely as the resistance $R$ (in ohms, Ω) of the conductor. If the current is 12 A when the resistance is 240 Ω, what is the current when the resistance is 540 Ω?
3. Hooke’s law states that the distance $(d_s)$ that a spring is stretched supporting a suspended object varies directly as the mass of the object $(m).$ If the distance stretched is 18 cm when the suspended mass is 3 kg, what is the distance when the suspended mass is 5 kg?
4. The volume $(V)$ of an ideal gas at a constant temperature varies inversely as the pressure $(P)$ exerted on it. If the volume of a gas is 200 cm3 under a pressure of 32 kg/cm2, what will be its volume under a pressure of 40 kg/cm2?
5. The number of aluminum cans $(c)$ used each year varies directly as the number of people $(p)$ using the cans. If 250 people use 60,000 cans in one year, how many cans are used each year in a city that has a population of 1,000,000?
6. The time $(t)$ required to do a masonry job varies inversely as the number of bricklayers $(b).$ If it takes 5 hours for 7 bricklayers to build a park wall, how much time should it take 10 bricklayers to complete the same job?
7. The wavelength of a radio signal (λ) varies inversely as its frequency $(f).$ A wave with a frequency of 1200 kilohertz has a length of 250 metres. What is the wavelength of a radio signal having a frequency of 60 kilohertz?
8. The number of kilograms of water $(w)$ in a human body is proportional to the mass of the body $(m).$ If a 96 kg person contains 64 kg of water, how many kilograms of water are in a 60 kg person?
9. The time $(t)$ required to drive a fixed distance $(d)$ varies inversely as the speed $(v).$ If it takes 5 hours at a speed of 80 km/h to drive a fixed distance, what speed is required to do the same trip in 4.2 hours?
10. The volume $(V)$ of a cone varies jointly as its height $(h)$ and the square of its radius $(r).$ If a cone with a height of 8 centimetres and a radius of 2 centimetres has a volume of 33.5 cm3, what is the volume of a cone with a height of 6 centimetres and a radius of 4 centimetres?
11. The centripetal force $(F_{\text{c}})$ acting on an object varies as the square of the speed $(v)$ and inversely to the radius $(r)$ of its path. If the centripetal force is 100 N when the object is travelling at 10 m/s in a path or radius of 0.5 m, what is the centripetal force when the object’s speed increases to 25 m/s and the path is now 1.0 m?
12. The maximum load $(L_{\text{max}})$ that a cylindrical column with a circular cross section can hold varies directly as the fourth power of the diameter $(d)$ and inversely as the square of the height $(h).$ If an 8.0 m column that is 2.0 m in diameter will support 64 tonnes, how many tonnes can be supported by a column 12.0 m high and 3.0 m in diameter?
13. The volume $(V)$ of gas varies directly as the temperature $(T)$ and inversely as the pressure $(P).$ If the volume is 225 cc when the temperature is 300 K and the pressure is 100 N/cm2, what is the volume when the temperature drops to 270 K and the pressure is 150 N/cm2?
14. The electrical resistance $(R)$ of a wire varies directly as its length $(l)$ and inversely as the square of its diameter $(d).$ A wire with a length of 5.0 m and a diameter of 0.25 cm has a resistance of 20 Ω. Find the electrical resistance in a 10.0 m long wire having twice the diameter.
15. The volume of wood in a tree $(V)$ varies directly as the height $(h)$ and the diameter $(d).$ If the volume of a tree is 377 m3 when the height is 30 m and the diameter is 2.0 m, what is the height of a tree having a volume of 225 m3 and a diameter of 1.75 m?