1
1
Linear equations
and functions
ALGEBRAIC MODELLING
In 1905, the German scientist Albert Einstein proposed a new theory of physics for small particles of matter (such as atoms) moving at very high speeds. Aged only 26 years, he hypothesised in his famous Theory of Special Relativity that matter (mass) could be converted to large amounts of energy. Einstein’s mass–energy relation was summarised by the now well-known formula
E = mc2
where E stands for energy, m for mass and c for the speed of light (300 000 000 m/s).
Einstein’s new model revolutionalised conventional laws of physics and led to the development of nuclear energy.
A criminologist studying the effects of increasing police patrols on the streets of a big city noticed the following pattern.
Is it possible for the criminologist, like Einstein, to find a general mathematical rule to fit this pattern?
Algebra is the branch of mathematics that uses symbols called pronumerals to represent
number patterns and relationships. Algebraic modelling is the study of patterns and relation-ships occurring in nature, industry and society and the formulation of a mathematical rule or ‘model’ to describe such relationships.
In this chapter you will learn how to:
simplify and expand algebraic expressions solve different types of linear equations apply formulas to solve a variety of problems graph linear functions on the number plane
understand the meaning of gradient and vertical intercept of a linear function examine real life situations that can be modelled by linear functions.
No. of police 50 150 250 300
ALGEBRAIC TERMS
An algebraic expression is a general statement involving pronumerals (also called
variables). Pronumerals are letters of the alphabet that stand for numbers. An algebraic
expression is made up of terms. For example, 3x2− x + 10 has 3 terms: 3x2, −x and 10.
Types of algebraic terms
The expression 2x3− x2+ 7x + 1 has 4 terms:
2x3 is a cubic term because it involves a power of 3 (the pronumeral ‘cubed’).
−x2 is a quadratic term because it involves a power of 2 (the pronumeral ‘squared’).
7x is a linear term because it involves a power of 1 (the pronumeral is raised to a power of 1).
1 is a constant term because it involves just a number and no pronumeral.
Adding and subtracting terms
Example 1
Simplify the following.
(a) 2a2− a + 5 + 8a (b) 4kr − 6pr − pr + 10kr (c) 2xy + 4 − y + 4yx
Solution
(a) 2a2− a + 5 + 8a = 2a2− a + 8a + 5 = 2a2+ 7a + 5
(b) 4kr − 6pr − pr + 10kr= 4kr + 10kr − 6pr − pr = 14kr − 7pr
(c) 2xy + 4 − y + 4yx = 2xy + 4yx − y + 4
= 6xy − y + 4
Algebraic terms are always written with the number first, followed by the pronumerals in alphabetical order: for example, 14kr, not 14rk or kr14.
Also, in algebraic expressions, terms with higher powers or more pronumerals are usually written first: for example, −3x2+ 4xy − y + 7, not −y − 3x2+ 7 − 4xy.
Multiplying terms
Example 2
Simplify the following.
(a) −3bc × 8ab (b) 4p2q2× (c) (−5r)2
Only like terms can be added or subtracted.
−a and 8a are like terms, but a2 and
a are not like terms.
kr and pr are not like terms.
xy and yx are like terms.
am× an= am+n
When multiplying terms with powers of the same base, add the powers.
---Solution
(a) −3bc × 8ab =−24ab2c (b) 4p2q2× = 6p5q3 (c) (−5r)2=−5r ×−5r = 25r2
Dividing terms
Example 3
Simplify the following.
(a) 8m3n2 ÷ 2m2 (b) −15w ÷ 3w2 (c)
Solution
(a) 8m3n2 ÷ 2m2= (b)−15w ÷ 3w2 =
= 4mn2 =−
(c) =
1. Simplify the following.
(a) 4p2− 2p + 8p − p2 (b) 7de + 8df − 2de + 2df (c) 10 − 4m2+ 6m2− 5
(d) 3r + 4ar − 2ar − 7r (e) 4x2− 2xy + 3x + 4xy (f) −5 + 7u − 10 − 3u
(g) 2k + 8 − k2− 4k (h) y2+ 5y − 3y − 15 (i) 4z + z + 8t − 4z
(j) 8t2+ 4w − 8t2+ 2w (k) −3c2+ 7c − 7c2+ 6c (l) 15x − 5 + 3x + 5 2. Simplify the following.
(a) 4ak × 4am (b) −3d × 8cd (c) t2w × 10tw
(d) (5x3)2 (e) 9n2× (f) 7k3p × 2k2p
(g) −6m ×−6n (h) 5e × (i) (2xy2)2
(j) a2y × y2 (k) 10ab ×−2ab (l) −3p ×−6pq
(m) x2× x3× x (n) −4n3× n3 (o) (−2k2)2
3. Simplify the following.
(a) (b) (c) −15e2 ÷ 3e2
(d) 10u2t3 ÷ 5ut2 (e) (f)
(g) (h) a2 ÷ a3 (i)
(j) (k) u ÷ 4u (l)
3p3q 2
---am ÷ an= = am−n
When dividing terms with powers of the same base, subtract the powers. am
an
---6ak3 20ak
---8m3n2 2m2
--- –15w
3w2
---5 w ----6ak3
20ak --- 3k2
10
---Exercise 1-01:
Algebraic terms
1 4
---2n2 3 ---2e2
5 ---1
2
---1 4
---4m2 2m
--- 14x2y
2xy ---3x x3
--- 27pr
9pq – ---12g2h
20gh
--- 10m4n3
10m2n ---4de
– 20e –
--- 7xz
---(m) (n) −5t ÷ −20t (o) 6wx2 ÷ 15w2
(p) (q) 6a4m2 ÷ −2a2m2 (r)
Equipment: A calendar from any year.
1. Choose a month from the calendar and draw a
box around any block of 9 numbers. Example:
2. Add any line of 3 numbers (row, column, diagonal) going through the centre of the box
(for example, 10, 18 and 26).
3. Add another line of 3 numbers going through the centre.
4. There are two more lines that go through the centre. Find their sums as well.
5. What do you notice? Why does this pattern work? Hint: Let the number in the top left
corner be x.
FORMULAS
A formula is an algebraic rule describing a relationship between pronumerals. For example, the volume of a cylinder has the formula
V =πr2h
where r represents the radius of the cylinder’s base and h is its height. This means that the volume of the cylinder is the product of π, the radius squared and the height.
Example 4
The compound interest formula
A = P(1 + r)n
shows the amount $A to which a principal $P will grow if invested for n years at an interest rate of r per annum, where r must be written as a fraction or decimal. Calculate the amount to which an investment of $4000 will grow if invested at 11% p.a. for 6 years.
32f3g2 –
4f2g ---24cd
8de
--- 20n3x
10n x2
---Investigation:
Calendar patterns
10 11 12
17 18 19
24 25 26
Just for the record
AL-JABR IS ARABIC
In the ninth century, the Arabic mathematician al-Khwarizmi wrote a book called Hisab al-jabr w’almuqabala, meaning ‘The science of equations’. The Arabic word al-jabr meant the process of adding the same amount to both sides of an equation; but when it was translated into Latin, it was changed to algebra and became the name of a whole branch of mathematics, not just equation solving.
Solution
P = 4000, r = 11% = , n = 6. A = P(1 + r)n
= 4000
= 4000(1.11)6 = 7481.6582 …
≈ $7481.66
The final amount is $7481.66.
Example 5
From the top of a tower of height h m, an observer can see a distance d km to the horizon, where d is given by the formula
d= 8
What distance (to the nearest kilometre) can be seen from the top of Sydney Tower, which has a height of 305 m?
Solution
d = 8
= 8
= 8
= 62.4819 …
≈ 62 km
A distance of about 62 km can be seen from the top of Sydney Tower.
1. Find the area of a circle with a radius of 4.07 cm correct to 2 decimal places if the area
formula is A =πr2.
2. A kettle is boiled, and the water temperature in °C after t minutes is given by the formula T= 18t + 28
Find the temperature of the water:
(a) after 4 minutes (b) after 1 minutes (c) at the start
3. The formula for converting Australian dollars ($A) to US dollars (US$) is
US= 0.6676A
Convert the following $A amounts to US$, correct to the nearest cent.
(a) $20.00 (b) $4.50 (c) $8.99
4. Heron’s formula for finding the area of a triangle with side lengths a, b and c is
A = where s = (a + b + c)
For a triangle with sides of length 7 cm, 3 cm and 6 cm, find the value of s and hence calculate its area correct to 2 decimal places.
11 100
---1 11
100 ---+
⎝ ⎠
⎛ ⎞6
h 5
---h 5 ---305
5 ---61
Exercise 1-02:
Formulas
1 2
---s ---s( –a)(s–b)(s–c) 1 2
---5. The angle sum S of a polygon (shape) with n sides is S = 180(n − 2)° Find the angle sum of a:
(a) quadrilateral (b) hexagon (c) decagon
6. If an object is moving with speed u m/s and acceleration a m/s2, then its final speed v m/s after time t seconds is given by the formula
v = u + at
Calculate the final speed of a car after 5 seconds if its speed now is 6 m/s and it is accelerating at 2 m/s2.
7. A mass of 3.33 × 10–28 kg of uranium is converted into energy during nuclear fission
(explosion) according to Einstein’s formula E = mc2
where energy E is measured in joules (J), m is in kg and c = 300 000 000 m/s is the speed of light. Show that the amount of energy released during fission is 2.997 × 10−11 J. 8. The number of matches m needed to make this pattern of triangles is
m = 2t + 1 where t is the number of triangles in the pattern.
How many matches are required to make:
(a) 8 triangles? (b) 40 triangles? (c) 150 triangles?
9. The volume of a sphere with radius r is V = πr3
Calculate correct to 2 decimal places the volume of a sphere with radius 14.5 cm.
10. The time it takes a swing to go back and forth once (in seconds) is
T = 2π
where l is the length of the swing (in metres) and g is the gravitational acceleration. Find T correct to 2 decimal places if l = 2.35 m and g = 9.80 m/s2.
11. Use the compound interest formula
A = P(1 + r)n
to calculate the amount to which a principal P = $3600 will grow if invested at r = 14% p.a. for n = 5 years. Remember that r must be written as a fraction or decimal, and write your answer correct to the nearest cent.
12. The maximum distance d m that a ball covers if thrown with velocity v m/s is
d =
where g is the gravitational acceleration. Find d if v = 11.5 m/s and g = 9.8 m/s2. 13. The body mass index (BMI) of an adult is
B =
where M is the mass in kg and h is the height in m. Patty is 1.7 m tall and weighs 60 kg. (a) Calculate her body mass index correct to 1 decimal place.
(b) If a BMI between 21 and 25 is an indication of good health, how can Patty improve her health?
1 3 5 7
6 4 2
. . .
4 3
---l g
---v2 g
---14. The formula for converting Fahrenheit temperatures (°F) to Celsius (°C) is
C = (F − 32)
Convert the following temperatures to Celsius, to the nearest degree.
(a) New York, 77°F (b) Los Angeles, 100°F
(c) Rio de Janeiro, 59°F (d) normal body temperature, 98.4°F
15. The formula for converting speeds from kilometres per hour (km/h) to metres per second
(m/s) is
M =
Convert 80 km/h to the nearest metres per second.
16. The distance (in kilometres) that an observer can see to the horizon from the top of a
structure of height h m is
d = 8
What distance can be seen from the top of Sydney Harbour Bridge, at a height of 125 m?
17. Jill earns a weekly wage of $450 plus a commission of 12% of the value of cosmetics
she sells in excess of $2000. Her total pay is given by the formula P = 450 + 0.12(V − 2000)
where V is the value of the cosmetics sold. How much will Jill earn for selling $2310 worth of cosmetics?
18. The braking distance (in metres) of a bicycle travelling at a speed of V m/s is
d =
Calculate the braking distance when the speed of the bicycle is 6 m/s.
19. The surface area of a cylinder of radius r and height h is S = 2πr(r + h)
Calculate the surface area (correct to 2 decimal places) of a cylinder with radius 3 cm and height 8 cm.
20. The velocity V m/s required for a rocket to escape the Earth’s gravitational pull is V =
where g = 9.8 m/s2 (the gravitational acceleration) and r = 6 378 000 m (the radius of the Earth). Show that the escape velocity of a rocket leaving the Earth’s atmosphere is approximately 11.2 km/s.
21. The simple interest earned when investing a principal $P at an interest rate r per annum
for n years is
I = Prn
where r must be written as a fraction or decimal. Calculate the interest earned by $2000 invested at 8.5% p.a. for 3 years.
22. The volume of a cone with radius r and height h is V = Ah
where A =πr2 is the area of the circular base. For a cone with radius 5 cm and height
6 cm, find correct to 2 decimal places:
(a) the area of the base (b) the volume of the cone
5 9
---5k 18
---h 5
---V ---V( +1) 2
---2gr
1 3
---23. The cost of hiring a limousine is $60 plus $2.70 for each kilometre travelled. This can
be expressed as
C = 60 + 2.7k
where k is the number of kilometres travelled. Calculate the cost of hiring a limousine to travel 35 km.
24. The formula for converting lengths measured in feet F to metres M is M = 0.3048F
Convert the following lengths to metres, correct to 2 decimal places.
(a) average height of a 12-year-old 5 feet
(b) height of a room 8.3 feet
(c) distance a kangaroo can hop in one leap 25 feet (d) length of an American football field 300 feet
(e) height of Mount Everest 29 028 feet
25. This formula gives the time that a planet takes (in seconds) to revolve around the Sun:
T = 2π
where r is distance between the planet and Sun (in metres), G = 6.67 × 10−11 Nm2/kg2 is
a gravitational constant, and M = 1.994 × 1030 kg is the mass of the Sun. Use this formula
to calculate the orbit times of the following planets, correct to the nearest day. (a) Mars: r = 2.28 × 1011 m
(b) Mercury: r = 5.830 × 1010 m
(c) Earth: r = 1.497 × 1011 m r3 GM
---Just for the record
T
HE TALLESTBUILDING(
S)
IN THE WORLDThe tallest building in the world was for many years the Sears Tower in Chicago, USA, built in 1973 with 110 storeys and a height of 443 m. However, in 1996 the Petronas Twin Towers of Kuala Lumpur, Malaysia were constructed, pipping this record with a height of 452 m and 88 storeys. These two office buildings are identical in shape and size, linked by a skybridge between levels 41 and 42. Both are of similar height to the Sears Tower, but gain their height advantage by having 74 m pinnacles (spires) built on top.
Here are the heights of some other famous buildings and structures:
CN Tower, Toronto 553 m (tallest structure in the world)
World Trade Centre, New York 417 m
Empire State Building, New York 381 m
Tokyo Tower 333 m
Eiffel Tower, Paris 320 m
Sydney Tower 305 m
Rialto Tower, Melbourne 253 m
Leaning Tower of Pisa 55 m
Use the formula d = 8 to calculate the distance that can be seen to the horizon from the top of the Petronas Twin Towers.
---When prescribing medicine for infants (babies), the amount of medicine to be given, called the dosage, is calculated by the formula
D=
where m is the age of the infant in months and A is the adult dosage. This formula is called
Fried’s rule.
1. Kathy is an 18-months-old infant. Use Fried’s rule to calculate her medicine dosage if
the adult dose for this medicine is 15 mL.
There are two other models or formulas for determining infant dosage: Young’s rule is D = where y is the age of the child in years.
Clark’s rule is D = where k is the mass of the child in kilograms.
2. Use Young’s rule to calculate Kathy’s dosage (correct to 2 decimal places) and compare
it with your answer in question 1.
3. Use Clark’s rule to calculate Kathy’s dosage (correct to 2 decimal places) if Kathy
weighs 8.5 kg.
4. Calculate the mean and range of the three answers above.
5. At what age does Fried’s model assume that adulthood is reached?
EXPANDING EXPRESSIONS
Algebraic expressions involving grouping symbols (brackets) can be expanded and simplified. Expanding means rewriting the expression ‘the long way’ and removing the grouping symbols.
Example 6
Expand:
(a) 2k(3k + 4n − 7) (b) −3(d2− 3d + 8)
Solution
(a) 2k(3k + 4n − 7) = 6k2+ 8kn − 14k (b) −3(d2− 3d + 8) =−3d2+ 9d − 24
Example 7
Expand and simplify:
(a) x(x − 1) + 4(x + 1) (b) 2m(m − 3) − (m − 6) (c) 4u2(2 − u) + u(u − 3)
Solution
(a) x(x − 1) + 4(x + 1)= x2− x + 4x + 4 = x2+ 3x + 4
(b) 2m(m − 3) − (m − 6)= 2m2− 6m − m + 6 = 2m2− 7m + 6
(c) 4u2(2 − u) + u(u − 3)= 8u2− 4u3+ u2− 3u =−4u3+ 9u2− 3u
Modelling activity:
Who rules?
mA 150
---1. Expand:
(a) 3p(p − ap) (b) −4(2k + 4) (c) −d(de − 5)
(d) k2(7m + k) (e) 9ab(a2− b2) (f) 2t(3u − 4t)
(g) −6n(4 − n) (h) 5x(x3+ x) (i) −(2a2− 4)
(j) 5b(a2+ 3b − 7) (k) −(x2− 4x + 10) (l) 3h(h − 7e − 4h2) (m) y(2y + 3 − y2) (n) de(d2− 2 + e2) (o) −v(2av + v − 2a) 2. (a) Evaluate 7 − 5 and 5 − 7. How are the two answers related?
(b) Evaluate 4 − 10.
(c) Is a − b the same as −(b − a)? Can you prove it?
3. Expand and simplify:
(a) 5(x + 4) − 2(x + 3) (b) 3(d − 4) − 2(d + 5) (c) 6(r + 10) − 4(r − 5) (d) 8( f + 2) − ( f + 7) (e) 3(2x − 4) − 5(3x + 4) (f) 6x(x + 4) − 3x(x − 1) (g) 3b(b + 5) − b(b − 8) (h) 4w(w − 7) − w(w + 1) (i) k(k + 3) + 3(k + 3) (j) b(4 − b) + 2(b + 5) (k) v(2v + 4) − 4(2v + 4) (l) 3t(t + 6) − 2(2t − 7) (m) e(3e + 5) − (2e − e2) (n) a(a + 3) + 7(a − 3) (o) p(p − q) − q(q − p) (p) 3(2x + 1) + 2x(x + 1)
Expanding can also be used with numerical expressions to make a calculation simpler. Using the rules of expanding, it is possible to multiply by a number close to 10, 100 or 1000 mentally or on paper, without using a calculator. Study the following examples.
1. 35 × 11= 35 × (10 + 1) Break 11 into 10 + 1. = 35 × 10 + 35 × 1 Expanding
= 350 + 35 Simplifying
= 385
2. 43 × 102= 43 × (100 + 2) Break 102 into 100 + 2. = 43 × 100 + 43 × 2 Expanding
= 4300 + 86 Simplifying
= 4386
3. 75 × 9= 75 × (10 − 1) Break 9 into 10 − 1. = 75 × 10 − 75 × 1 Expanding
= 750 − 75 Simplifying
= 675
Use expansion to evaluate these expressions without using a calculator.
(a) 25 × 12 (b) 18 × 9
(c) 6 × 105 (d) 87 × 11
(e) 50 × 99 (f) 32 × 8
The rule a(b + c) = ab + ac is called the distributive law because it distributes a difficult multiplication across two simpler multiplications, which are then either added or subtracted.
Exercise 1-03:
Expanding expressions
SOLVING EQUATIONS
An equation contains an algebraic expression and an equals (=) sign. For example, 3x − 4 is an expression, but 3x − 4 =−13 is an equation. To solve an equation, we find the value of the pronumeral that makes the equation true. The process of solving an equation requires the use of inverse (opposite) operations.
Example 8
Solve these equations.
(a) 3p + 4 = 12 − p (b) = 5
(c) 4(1 − 2t) = 20 (d) 5(4k + 6) = 2k
Solution
(a) 3p + 4 = 12 − p
4p + 4 = 12 Adding p to both sides 4p = 8 Subtracting 4 from both sides
p= Dividing both sides by 4
= 2
We can check the answer by substituting p = 2 back into the equation. LHS means ‘left hand side’. RHS means ‘right hand side’.
LHS= 3p + 4 RHS= 12 − p
= 3(2) + 4 = 12 − 2
= 10 = 10
= LHS
(b) = 5
2m − 7= 15 Multiplying both sides by 3 2m= 22 Adding 7 to both sides
m= Dividing both sides by 2
= 11 Checking:
LHS =
=
= = 5
= RHS To solve an equation:
1. Perform inverse operations on both sides of the equation.
2. Aim to have the pronumeral on one side and a number on the other side. Example: x = 12
2m–7 3
---8 4
---2m–7 3
---22 2
---2 11( )–7 3 ---22–7
3 ---15
---(c) 4(1 − 2t)= 20
4 − 8t= 20 Expanding the LHS
−8t= 16 Subtracting 4 from both sides t= Dividing both sides by −8
=−2 Checking:
LHS = 4[1 − 2(−2)]
= 4(1 + 4)
= 4(5)
= 20 = RHS (d) 5(4k + 6)= 2k
20k + 30= 2k Expanding the LHS
18k + 30= 0 Subtracting 2k from both sides 18k=−30 Subtracting 30 from both sides
k= Dividing both sides by 18
=− (or −1 ) Checking:
LHS = 5[4(− ) + 6] RHS = 2(− )
= 5(− ) =−
=− = LHS
Solve these equations.
1. 3d + 2 = 20 2. 2p − 3 = 2
3. 2u + 6 = 2 4. 5k + 3 − 2k = 42
5. 5a + 3 =−12 6. 12x + 8 = 4
7. 3 − 2a =−6 8. 3d = d − 1
9. 8x = 2(x − 6) 10. = 10
11. = 9 12. − = 8
13. 3(x + 2) = 45 14. 2(3a + 4) = 20
15. 4(2x − 9) =−12 16. 6(2 − 3b) =−24
17. −4(k + 1) = 16 18. 3(2c − 1) + 4c = 10
19. = 4 20. = 1
21. = 3 22. 3u + 7 = 2u − 10
23. 2y − 4 = 26 − 3y 24. 4k + 18 = k − 18 16
8 –
---30 –
18
---5 3
--- 2
3
---5 3
--- 5
3
---2 3
--- 10
3
---10 3
---Exercise 1-04:
Solving equations
2f –4 2 ---3x
4
--- 2x
5
---3k–7 2
--- y+2
3 – ---2m–1
---25. 9f + 3 = 4f − 22 26. 5(2a + 5) = 8a − 1
27. 2(3 − a) = 4 28. 11 − 4u = 19
29. 7(3 − 3y) = 6(y + 6) 30. 3(4 − 3x) = 2(3 − 2x)
EQUATIONS INVOLVING ALGEBRAIC FRACTIONS
For equations involving algebraic fractions, both sides can be multiplied by a common
multiple of the denominators, found by multiplying the denominators together. This way, we
convert all fractions into whole numbers and then solve the equation the usual way.
Example 9
Solve these equations.
(a) − = 5 (b) − =−3
(c) = (d) − 1 = + 4
Solution
(a) − = 5 Common multiple is 4 × 2 = 8.
8 = 8(5)
8 − 8 = 40
2(3a) − 4(a)= 40 6a − 4a= 40 2a= 40 a= 20 Checking:
LHS= −
= − 10
= 15 − 10
= 5 = RHS
(b) − =−3 Common multiple is 10 × 3 = 30.
30 = 30(−3)
30 − 30 =−90
3(3h − 1) − 10(2h + 1) =−90 9h − 3 − 20h − 10 =−90
−11h − 13 =−90
−11h =−77 h = 7 3a
4 --- a
2
--- 3h–1
10
--- 2h+1 3 ---4x–2
6 --- x 4 --- a 5 --- 3a 2 ---3a 4 --- a 2 ---3a 4 --- a 2 ---– ⎝ ⎠ ⎛ ⎞ 2 3a 4 ---⎝ ⎠ ⎛ ⎞ 1 4 a 2 ---⎝ ⎠ ⎛ ⎞ 1
3 20( ) 4 --- 20 2 ---60 4
---3h–1 10
--- 2h+1 3 ---3h–1
10
--- 2h+1 3 ---–
⎝ ⎠
⎛ ⎞
3 3h–1 10
---⎝ ⎠
⎛ ⎞
1
(c) =
4(4x − 2)= 6x Same as multiplying both sides by 24: see below 16x − 8= 6x
10x − 8= 0 10x= 8
x= =
When both sides of an equation are single fractions, a shortcut to multiplying by a common denominator is to cross-multiply.
(d) − 1= + 4
10 = 10
− 10= + 40
2a − 10= 15a + 40
−13a − 10= 40
−13a= 50
a= (or −3 )
Solve these equations.
1. − = 7 2. + = 14
3. + = 2 4. + = 10
5. + = 8 6. − = 7
7. = 8. =
9. − = 10. + 2 = − p
11. = + 12. = +
13. = − 14. − =
15. − = 5 16. =
17. + = 9 18. − = 1
19. − = 4 20. + = 5
4x–2 6 --- x 4 ---8 10 --- 4 5
---4x–2 6 --- x 4 ---a 5 --- 3a 2 ---a 5 ---–1
⎝ ⎠
⎛ ⎞ 3a
2 ---+4
⎝ ⎠ ⎛ ⎞ 10a 5 --- 30a 2 ---50 13 – --- 11 13
---Exercise 1-05:
Equations involving algebraic fractions
x 5 --- x 8 --- m 3 ---- m 5 ----4n 5 --- 2n 6 --- 2a 3 --- a 4 ---2t 3 --- 4t 5 --- 3r 4 --- 2r 5 ---4x 3
--- 2x–4 10
--- 3n
5
--- n+7 4 ---x 3 --- 1 5 --- 1 4 --- p 2 --- 4 3 ---3 5 --- r 2 --- 7 10 --- 3u 4 --- 2u 5 --- 1 4 ---7 10 --- b 4 --- 2 5 --- z 4 --- 1 5 --- 3 4 ---k+6
5
--- k–2 3 --- r 12 --- 3 5 ---h+4
4 --- h
3
--- a+3
6
--- 2a–4 5 ---4m+1
5
--- 3m+1 10
--- p+4
5
---21. − = 7 22. − = + 4
23. − = 1 24. = 3 − w
25. − = 3 26. =
27. = 28. = +
29. = 30. − 5 = +
EQUATIONS AND FORMULAS
Sometimes when solving a problem involving a formula, the answer is not immediately found after substituting in a value. Instead an equation results, which must then be solved.
Example 10
The formula for converting temperatures from the Fahrenheit scale to the Celsius scale is C = (F − 32)
Use the formula to convert a temperature of 100°C to °F.
Solution
When C = 100, 100 = (F − 32) 900 = 5(F − 32)
= 5F − 160 1060 = 5F
F =
= 212 Therefore, 100°C = 212°F.
6n–7 3 --- n
8
--- 4k
3 --- 1
2 --- k
4 ---2d
3 --- 5d
10
--- 5w+4
3 ---2y–1
2
--- 3y–4 5
--- 4r–2
6
--- r+7 4 ---2b+4
3
--- 5b–1 2
--- 5
6 --- u
2 --- 2
3 ---4p
3 --- 8
10
--- 3h
4
--- h
5 --- 1
2
---Study tips
USING A MATHS JOURNAL OR SUMMARY BOOK
Buy a small exercise book or notebook to use as your Maths journal or summary book, sometimes called a theory book or formulas book. Alternatively, some students like to use a looseleaf folder so that they can keep rewriting and revising their study notes. A Maths journal is for summarising each topic—listing key facts, formulas and examples. It can also be a personal diary or record of your comments and reflections about your mathematical learning. Studies show that more effective learning occurs when you restructure and re-explain difficult concepts and ideas in your own words.
Each chapter in this book has sections that refer to your Maths journal. Important facts and formulas are boxed and labelled by a (record) button, so that you can include them in your summaries. The Topic summary and Your say in the Chapter reviews contain questions and ideas for organising your notes and thinking about your Maths study.
5 9
---5 9
---Example 11
The surface area of a rectangular prism of length l, breadth b and height h is given by
S = 2lb + 2lh + 2bh
If a rectangular prism with surface area 132 cm2 has
length 7 cm and breadth 3 cm, find its height.
Solution
S = 132, l = 7, b = 3
132 = (2 × 7 × 3) + (2 × 7 × h) + (2 × 3 × h)
= 42 + 14h + 6h
= 42 + 20h 90 = 20h
h =
= 4
The height of the rectangular prism is 4 cm.
Example 12
Young’s rule for finding the medicine dosage for infants is D =
where y is the age of the infant in years and A is the adult dose. How old is Liam if his medicine dose is 3.6 mL and the adult dosage is 18 mL?
Solution
D = 3.6, A = 18
3.6 = 3.6(y + 12) = 18y 3.6y + 43.2 = 18y 43.2 = 14.4y
y = = 3 Liam is 3 years old.
1. The number of matchsticks needed to make a pattern of s squares is m = 3s + 1
(a) How many matches are needed to make:
(i) 4 squares? (ii) 10 squares?
(b) How many squares can be made from:
(i) 22 matches? (ii) 55 matches?
l
h
b
90 20 ---1 2
---1 2
---yA y+12
---18y y+12
---43.2 14.4
---Exercise 1-06:
Equations and formulas
1 2 3 4 5
2. Find the base length of a triangle that has an area of 90 cm2
and a perpendicular height of 15 cm, given the area formula A = bh
3. The average speed of a moving object in metres per second is
S =
where d is the distance travelled in metres and t is the time taken in seconds. (a) Find the distance travelled by a car in 20 seconds if its speed is 15 m/s.
(b) Find the time taken (to the nearest second) for a cyclist to travel 250 m if his speed is 4.2 m/s.
4. The formula for the area of a trapezium is
A = (x + y)h
where x and y are the lengths of the parallel sides, and h is the distance between them. What is the length of one parallel side of a trapezium if the other parallel side is 7 m, the distance between them is 5 m and the area is 22.5 m2?
5. The angle sum of a polygon with n sides is S = 180(n − 2)°. (a) Calculate the angle sum of:
(i) a triangle (ii) a pentagon
(b) Which shape has an angle sum of:
(i) 1080°? (ii) 720°?
6. The mean of three numbers x, y and z is
M =
If three numbers have a mean of 17, and two of the numbers are 10 and 20, find the third number.
7. The circumference of a circle with radius r is C = 2πr
If a circle has a circumference of 50.27 cm, find its radius to the nearest centimetre.
8. The compound interest formula
A = P(1 + r)n
shows the amount ($A) to which a principal ($P) will grow if invested at a rate of r per annum over n years, where r must be written as a fraction or decimal. How much principal needs to be invested at 11% p.a. for it to grow to $6000 in 3 years? Express your answer to the nearest cent.
9. A kettle is boiled, and the temperature (°C) of the water after t minutes is T = 18t + 28
After how many minutes is the temperature:
(a) 64°C? (b) 92.8°C?
10. The formula for converting miles to kilometres is K = 1.61M
where M is the distance in miles. Use this formula to convert the following distances to miles, correct to 2 decimal places.
(a) 5 km (b) 1.5 km
b
15 cm
1 2
---d t
---1 2
---11. The number of chairs c that can be seated around t square tables is c = 2t + 2.
(a) How many chairs can be seated around:
(i) 5 tables? (ii) 12 tables?
(b) How many tables are required to seat:
(i) 24 chairs? (ii) 40 chairs?
12. Given the formula E = mc2, where m is in kilograms, E is in joules, and c m/s is the speed
of light, find m to 3 significant figures if c = 3 × 108 m/s and E = 1.45 × 10–11 J. 13. Jill earns a weekly wage of $450 plus a commission of 12% on the value of cosmetics
she sells in excess of $2000. Her total pay is given by the formula P = 450 + 0.12(V − 2000)
where V is the value of the cosmetics sold. What was the value of the cosmetics Jill sold if her total pay was $488.40?
14. The surface area of a rectangular prism of length l, breadth b and height h is SA = 2lb + 2lh + 2bh
If a rectangular prism has length 4 cm, breadth 2 cm and surface area 58 cm2, find its height.
15. The formula for converting kilometres k to miles M is
M =
Use this formula to convert 12 miles to kilometres.
16. The number of matchsticks needed to make this pattern of triangles is m = 2t + 1
where t is the number of triangles in the pattern.
How many triangles can be made with:
(a) 37 matches? (b) 55 matches?
17. The size of each angle (in degrees) in a regular polygon with n sides
is given by the formula
a = 180 −
(a) Find the size of one angle in a regular octagon. (b) Find the number of sides in a regular polygon if
one angle is 162°.
18. A babysitter charges $12 for the first hour and $8 for each hour after that. This can be
expressed by
C = 12 + 8(h − 1)
where h is the number of hours worked. If the babysitter earned $36 for one night, how many hours did he work?
19. The formula for converting Australian dollars ($A) to US dollars (US$) is US = 0.6676A. Convert the following US amounts to Australian dollars, correct to the nearest cent.
(a) US$1.00 (b) US$20.00 (c) US$7.50
1 1 2 1 2 3
Table Chair
5k 8
---1 3 5 7
6 4 2
. . .
a°
---20. Julie works in a high-rise office building on the 32nd floor. At lunchtime, she gets into
a lift which travels downwards at a speed of 2 floors per second. This is described by the formula F = 32 − 2t where F is the floor being passed and t is the time in seconds. (a) Through which floor is Julie passing after:
(i) 8 seconds? (ii) 14 seconds?
(b) After how many seconds will Julie pass: (i) the 18th floor? (ii) the 8th floor?
21. According to one theory, the recommended nightly hours of sleep for a child is
S = 8 +
where a is the age of the child in years. How old is a child who requires 14 hours of sleep each night?
22. If an object travels with initial velocity u m/s and acceleration a m/s2, then after time t seconds its final velocity (in metres per second) will be
v = u + at
Find the acceleration of an object if its initial velocity is 8 m/s and its final velocity after 12 seconds is 44 m/s.
23. The volume of a cone with radius r and height h is V = πr2h
If a cone has volume 256 cm3 and radius 7 cm, find its height correct to 2 decimal places. 24. The cost of hiring a limousine is
C = 60 + 2.7k
where C is in dollars and k is the number of kilometres travelled. Find the kilometres travelled if the total cost amounted to $232.80.
25. The equation of a straight line is y = mx + b
where m is the gradient of the line, b is the y-intercept and (x, y) is any point on the line. Find the gradient of the line that passes through (5, 6) and has a y-intercept of −4.
26. Nick weighs 79 kg and has a body mass index (BMI) of 24.5. Calculate his height to the
nearest centimetre if the BMI formula is B =
where M is the person’s mass in kilograms and h is their height in metres.
LINEAR FUNCTIONS
Example 13
(a) Complete this table of values for the rule y = 3x − 2.
(b) What pattern do you notice along the bottom row of y-values? (c) Graph y = 3x − 2 on a number plane.
(d) Find the gradient and y-intercept of the graph.
x −1 0 1 2 3
y
18–a 2
---1 2
---1 3
---Solution
(a) Substituting x-values to find y:
(b) y-values increase by 3.
(c) Graphing the points (−1,−5), (0,−2), (1, 1), (2, 4), (3, 7) from the table:
(d) From the graph, gradient = = = 3, and the y-intercept is −2. The y-intercept or vertical intercept is where the line crosses the y-axis.
The meaning of linear function
y = 3x − 2 is called a linear function because its graph is a straight line.
A linear function has the form y = mx + b, where m and b are constants (numbers). A linear function has 2 terms: mx is called the linear term as it contains the variable x (but
not raised to a power), while b is called the constant term because it is just a number (no variable).
Functions involving higher powers of x such as x2 and x3 are called non-linear functions
because their graphs are not lines, but curves.
x −1 0 1 2 3
y −5 −2 1 4 7
y
x 7
6 5 4 3 2 1
0
−1
−2
−3
−4
−5
−2 2 3 4
y-intercept 1
3 1
3 y = 3x − 2
rise 3
run 1 5
4
3
2
1
0 1 2 3
−1
rise run --- 3
1
---The gradient of a line is its steepness, given by the formula
m = rise( )↑
run( )→
---Run
The gradient and y-intercept of y
=
mx
+
b
When a linear function is written in the form y = mx + b, its gradient and y-intercept are easily identified. The gradient is m and the y-intercept is b.
For example, for the gradient m is 3 and the y-intercept b is −2.
Example 14
Write the gradient and y-intercept of y = 2x + 1 and hence graph the linear function.
Solution
Gradient = 2, y-intercept = 1. To graph the line:
Step 1 Plot the y-intercept 1 on the y-axis.
Step 2 From this point, make a gradient of 2 by moving across 1 unit and up 2 units and marking the point. Step 3 Rule a line through this point and the y-intercept. Step 4 Label the line with the equation.
Example 15
(a) Complete this table of values for the rule y =−2x + 4.
(b) What pattern do you notice along the bottom row of y-values? (c) Graph y =−2x + 4.
(d) Write its gradient and y-intercept.
Solution
(b) y-values decrease by 2. (c)
(d) From the equation or graph, gradient =−2, y-intercept = 4.
x −1 0 1 2 3
y
(a) x −1 0 1 2 3
y 6 4 2 0 −2 y-intercept
gradient
y = 3x − 2
1 2 3 4 5
4 3 2 1
0 y
−1
−2
2
1
x y = 2x + 1
−1
1 2 3 4 5
4 3 2 1
0 y
−1
−2
−2 1
x
y =−2x + 4
−2 6
Positive and negative gradients
While a positive gradient means that the graph is sloping upwards (from left to right) and the y-values are increasing, a negative gradient means the line is sloping downwards and the y-values are decreasing. We can think of a negative gradient as involving a negative ‘rise’, which is a ‘fall’.
A gradient of −2 means that as the x-values increase by 1, the y-values decrease by 2. To draw a gradient of −2 on the number plane, move across 1 unit (as usual), but then go down 2 units.
Example 16
Write the gradient and y-intercept of y = x − 1 and graph the line.
Solution
Gradient = , y-intercept =−1. To graph the line:
Step 1 Plot the y-intercept −1.
Step 2 To make a gradient of , move across 3 units and up 2 units and mark the point.
Step 3 Rule a line through this point and the y-intercept.
Step 4 Label the line with the equation.
Gradient Graph
m is positive
m is negative
m is a fraction
Graph sloping downwards Graph sloping upwards
Positive gradient
y
x
Negative gradient
y
x
2 3
---1 2 4
3 2 1
0 y
−1
−2
2
3
x
−2
−3 5
−3
y = x 23 − 1
−1
2 3
---2 3
---Remember, m = rise run
---m
1
m
1
a b
-- a
b
What does a line with a gradient of 0 look like? What does a line with a gradient of 1 look like? What is the highest gradient a line can have?
1. Write the equations of the lines with the following gradients and y-intercepts, in the form y = mx + b.
2. Write the gradient and y-intercept of each of these lines, and hence write their equations.
Gradient y-intercept
(a) 3 7
(b) −2 1
(c) 1 −1
(d) −
(e) − 0
(f) 0 5
Think:
Zero gradient
Exercise 1-07:
Linear functions
1 3
--- 1
2
---5 4
---1 2 3
2 1
0 y
−1
−2 x
−2
−3 4
1 2 3
2 1
0 y
−1
−2 x
−2
−3 4
−4 1 2
3 2 1
0 y
−1
−2 x
−2 4 5
−1 −1
−1
1 2 3
2 1
0 y
−1
−2 x
−2 4
−1 1 2
3 2 1
0 y
−1
−2 x
−2 4
−1
1 2 2
1
0 y
−1
−2 x
−2
−1
(a) (b) (c)
3. Graph these linear functions.
(a) y = 4x − 3 (b) y = x + 2 (c) y = x − 2
(d) y =−2x (e) y = 2x − 5 (f) y = x − 1
(g) y =−x + 4 (h) y = x + 1 (i) y =− x + 3
This is the graph of a horizontal line:
1. What is its gradient?
2. What is its vertical intercept?
3. Complete the table of values for this line.
4. What happens to the y-values as the x-values
increase by 1?
5. What is the equation of this line?
1 2 1
0 y
−1 x
−2 −3 1 2 2 1 0 y
−1 x
−2 −3 1 2 3 2 1 0 y −1
−2 x
−2 −1 −1 −1 1 2 3 2 1 0 y −1
−2 x
−2 4 −1 1 2 3 2 1 0 y x −1 1 2 2 1 0 y −1
−2 x
−2 −1 −4 3 3 3 3 4 4 5
(g) (h) (i)
(j) (k) (l)
4 1 2 ---3 5 ---4 3 --- 1 4
---Just for the record
TRUCKS USE LOW GEARS
Australia is a fairly flat continent, but in Europe road signs often display the steepness of a road for drivers (or hill for hikers).
For example, means that the steepness is , also written 1 : 5.8 or 1 in 5.8. Bulli Pass, north of Wollongong in New South Wales, has a gradient of and Victoria Pass, east of Lithgow, has a gradient of . Which pass is steeper?
5.8 1 1 5.8 ---1 6 ---1 8
---Investigation:
Flatliners
1 2 1
0 y
−1
−2 −1 x
−3 3
x −1 0 1 2 3
A graphics calculator has a tall screen for displaying graphs of functions on a number plane. A function can be entered using its equation or a table of values.
Learn how to use a graphics calculator to graph the following linear functions.
1. y = 2x − 4 2. y = −x
3. y = x + 5 4. y = −1
Find out more about the following features of a graphics calculator: specifying the scale and range of the x- and y-values
zoom: magnifying a part of the graph
trace: displaying the coordinates of points on the graph.
A graphics calculator can also:
draw statistical graphs (for example, frequency histogram, pie chart) perform statistical calculations (for example, mean, standard deviation) solve equations
perform financial calculations (for example, compound interest).
THE GRADIENT AS A RATE OF CHANGE
Not only is the gradient a measure of the steepness of a line; it also shows how quickly the y-values are changing. Look at this table of values for y = 3x − 2.
Note that as the x-values increase by 1, the y-values increase by 3. On the graph of this function, for every ‘run’ of 1 unit, there is a ‘rise’ of 3 units.
The higher the gradient, the steeper the line, and the faster y changes relative to x.
Example 17
Find the equation of the linear function represented by this table of values.
Solution
As x-values increase by 1 unit, y-values increase by 5 units. Gradient m = 5. y-intercept =−4 (y-value when x = 0).
The equation is y = 5x − 4.
x −1 0 1 2 3
y −5 −2 1 4 7
x −1 0 1 2 3
y −9 −4 1 6 11
Technology:
The graphics calculator
1 3
---+ 1 + 1 + 1 + 1
+ 3 + 3 + 3 + 3
The gradient of a function is the rate of change of y.
+ 1 + 1 + 1 + 1
Horizontal and vertical lines
Horizontal lines have a ‘run’ but no ‘rise’.Since m = , the line has a zero gradient and has an equation of the form y = b where b is a constant. For example, y =−4 is the horizontal line drawn at right. A graph with a line showing zero gradient represents y-values that do not change (neither increasing nor decreasing). That is why y = b is also called a constant
function.
Vertical lines have a ‘rise’ but no ‘run’.
Since m = , the line’s gradient is not defined or is ‘infinite’ (a division by zero is undefined).
A vertical line has the form x = c where c is a constant. For example, x = 2 is the vertical line drawn at right.
1. Match these linear functions to their graphs below.
(a) y =−x (b) y = 2x + 5 (c) y = x − 2
(d) y =− x − 1 (e) y = 4 − 2x (f) y = x + 2
2. Write the gradient and y-intercept for each of these tables of values and hence write their
equations.
(a) x −1 0 1 2 (b) x −1 0 1 2
y 2 4 6 8 y −7 −2 3 8
1 2 1
0 y
−1
−2 −1 x
−3 3
−2
−3
4
−4
y = −4
0 run
---1 2 2
1
0 y
−1 x
−2
−1 3
−2
y = 2
rise 0
---Zero gradient
y
x Infinite
gradient
y
x
Exercise 1-08:
The gradient as a rate of change
1 2
---3 2
---A y B C
x
y
x
y
x
D y E F
x
y
x
y
3. What is the difference between a line with a gradient of 7 and a line with a gradient of ?
Illustrate with a diagram.
4. This is a matchstick pattern of houses.
1 house uses 6 matches. 2 houses use 11 matches. 3 houses use 16 matches. (a) Complete this table, where h = number of houses and m = number of matches.
(b) Find the linear function (formula) for m in terms of h. (c) How many matches will be required to make:
(i) 20 houses? (ii) 40 houses? (iii) 100 houses?
(d) If this function is graphed on the number plane, what will be its gradient and vertical intercept?
5. Graph the following lines.
(a) y =−3 (b) y = 1 (c) x = 4
(d) x =−3 (e) y = 0 (f) x =−
6. What is the difference between a line with a gradient of and a line with a gradient
of − ? Illustrate with a diagram.
The gradient of a line can also be described by the angle θ it makes with the x-axis.
There is a simple relationship between θ and the gradient m. Using trigonometry,
tan θ =
= = m
Therefore, the gradient is the same as the tangent ratio of the angle θ.
(c) x −1 0 1 2 (d) x −1 0 1 2
y −4 −3 −2 −1 y 3 10 17 24
(e) x −1 0 1 2 (f) x −1 0 1 2
y 0 −2 −4 −6 y −4 0 4 8
(h) x −1 0 1 2
y −1 −1 − 0
h 1 2 3 4 5 6
m
1 2
--- 1
2
---1 7
---1 2
---1 2
---1 2
---1 2
---Investigation:
Why tan
θ
=
m
y
x Run
Rise
θ opposite
adjacent ---rise run
---(g) x −1 0 1 2
The diagram below is like a protractor, showing different lines for values of θ between 0° and 180°. The table lists the gradients m = tan θ (to 4 decimal places) for each value of θ. You can verify some values using a calculator.
1. What do you notice about a line that makes an angle of 0° or 180° with the positive x-axis?
2. What do you notice about a line that makes:
(a) an angle of 45°? (b) an angle of 135°?
3. Why is tan 90° undefined?
4. Investigate what happens to tan θ as θ gets closer to 90° (for example, tan 85°, tan 89°, tan 89.999°).
5. What do you notice about the sign of tan θ when θ is more than 90°?
6. Comment on the symmetry of the gradient values of m in the table.
θ m = tan θ θ m = tan θ θ m = tan θ
0° 0 65° 2.1445 130° −1.1918
5° 0.0875 70° 2.7475 135° −1
10° 0.1763 75° 3.7321 140° −0.8391
15° 0.2679 80° 5.6713 145° −0.7002
20° 0.3640 85° 11.4301 150° −0.5774
25° 0.4663 90° undefined 155° −0.4663
30° 0.5774 95° −11.4301 160° −0.3640
35° 0.7002 100° −5.6713 165° −0.2679
40° 0.8391 105° −3.7321 170° −0.1763
45° 1 110° −2.7475 175° −0.0875
50° 1.1918 115° −2.1445 180° 0
55° 1.4281 120° −1.7321
60° 1.7321 125° −1.4281
90°
180° 0°
x 175°
170° 165°
160° 155°
150° 145°
140°
135° 130°125°
120°115°
110°105°100°
95° 85° 80°75° 70°
65° 60°
55°
50° 45°
40° 35°
30° 25°
20° 15°
LINEAR MODELLING
A function like y = 4x − 7 can be thought of as a ‘number machine’ that changes an input value, x, into an output value, y.
The graph of a linear function y = mx + b is a straight line, demonstrating that the variable y is changing (increasing or decreasing) at a steady rate. In fact, as x increases by 1 unit, y increases (or decreases) by m units.
Because the value of y depends on the value of x, y is called the dependent variable and x is called the independent variable. When graphed, the
independent variable is represented on the horizontal axis while the dependent variable is represented on the vertical axis.
Example 18
Find the linear function y = mx + b represented by this table of values:
Solution
The gradient and y-intercept are harder to identify here because the x-values are not consecutive (increasing by 1) and the y-value corresponding to x = 0 is not shown.
To find the gradient, choose any two points from the table and calculate their difference for both the x- and y-values. For example, choosing the first two points (1, 9) and (3, 17):
Gradient m =
=
=
= 4
To find the vertical intercept, write the formula y = mx + b, substitute m = 4 and choose any point in the table for (x, y), say (1, 9):
9 = 4(1) + b
= 4 + b b = 5
Therefore, the linear function is y = 4x + 5.
x 1 3 6 10 15
y 9 17 29 45 65
x 1 3 6 10 15
y 9 17 29 45 65
FUNCTION 13
5
x Input Process Output y
Independent variable Dependent variable
Dependent variable
Independent variable
+ 2
+ 8
rise run ---change in y change in x ---8
2
Linear models
When scientists and researchers observe number patterns occurring in nature and society, they try to find or fit a mathematical formula to represent the relationship. This is called
algebraic modelling. A model approximates real life phenomena algebraically and can be
described using a formula, a table of values or a number plane graph.
If the observed number pattern suggests a linear relationship, then we use the linear function y = mx + b to model the situation. This is called a linear model.
Example 19
Countries using the metric system measure temperature by the Celsius scale (°C), but the USA is still using the Fahrenheit scale (°F). The table lists some equivalent temperatures expressed in both scales. For example, 20°C is the same as 68°F.
(a) For this table of values, which variable (C or F) is the dependent variable?
(b) Show that a linear relationship exists between C and F. Find the equation of the function. (c) Graph this function.
(d) What does the vertical intercept of this graph represent? (e) What does the gradient represent?
(f) Use the graph to convert:
(i) 18°C to °F (ii) 100°F to °C
Solution
(a) F is the dependent variable (dependent on C).
As we are using C and F in place of the variables x and y, the linear function is F = C + b
To find b, substitute a point, say (5, 41): 41 = (5) + b
= 9 + b 32 = b So the linear function is F = C + 32.
C 5 10 15 20 25
F 41 50 59 68 77
(b) C 5 10 15 20 25
F 41 50 59 68 77
Formulas for the gradient:
m = =
m = =
change iny
change inx
--- change in dependent variable change in independent variable
---rise run
--- vertical change in position horizontal change in position
---+ 5 + 5 + 5
+ 9 + 9 + 9
m= (or 1.8)9 5
---9 5
---9 5
---9 5
---(c)
(d) The vertical intercept 32 means that 0°C is the same as 32°F.
(e) The gradient means that as C increases by 1, F increases by (or 1 ). (Or put another way, as C increases by 5, F increases by 9.)
(f) (i) 18°C ≈ 64°F (ii) 100°F ≈ 38°C
Example 20
A criminologist studying crime in a major city found that the number of crimes committed per week, C, decreased as the number of police patrolling the city, P, increased. She graphed her data and found a linear relationship. For example, with 150 police on patrol, the number of crimes per week was 2800.
(a) What is the independent variable? (b) What is the gradient of this linear
relationship?
(c) What does the gradient represent? (d) What is the vertical intercept? (e) What is the formula for C? (f) Use the formula to find how
many crimes were committed per week when:
(i) 100 police were on patrol (ii) 450 police were on patrol
Solution
(a) P, the number of police patrolling
(b) Using the first two points m =
= =−3
Gradient is negative because the function is decreasing.
(c) The gradient represents the reduction in crime for every new police officer. As P increases by 1, C decreases by 3. For every new officer, crimes per week decreased by 3.
5 10 15 20 25 30 35 40 45 50 0
18° 38°
C F
64°
100 90 80 70 60 50 40 30 20 10
F = C 9 + 32
5
Temperature conversion
9 5
--- 9
5
--- 4
5
---50 100 150 200 250
Police patrolling, P
Crimes per week,
C
3500 3000 2500 2000 1500 1000 500
300 0
(50, 3100)
(150, 2800)
(250, 2500)
(300, 2350)
Crime rate
+ 100
− 300
P 50 150
C 3100 2800
change in C change in P
---300 –
---(d) The formula is C = mP + b where b is the vertical intercept. Substituting m =−3 and the point (50, 3100),
3100=−3(50) + b
=−150 + b 3250= b
The vertical intercept is 3250 crimes per week. (e) C =−3P + 3250
(f) (i) C =−3(100) + 3250
=−300 + 3250
= 2950 crimes per week (ii) C =−3(450) + 3250
=−1350 + 3250
= 1900 crimes per week
1. Read the definition of a linear function on page 20, then state which of the following
functions are not linear.
A. y = x2− x +1 B. y = 4x −2 C. y =−2x − 1
D. y = x + 3x3 E. y =−x F. xy = 5
G. y = H. y = x − I. x + y = 5
2. Find the linear function for each of these tables of values.
3. The table shows the cost C of a phone call under the OzExpress Mobile Budget Plan, for
different call durations t.
(a) Which is the dependent variable, and what does it represent?
(b) If this relationship is graphed, which variable will be shown on the vertical axis? (c) Graph this relationship.
(d) What is the gradient, and what does it represent?
(e) What is the vertical intercept, and what does it represent? (f) Write the formula for C.
(a) x 3 10 12 19 (b) x 7 15 23 30
y 34 97 115 178 y 165 125 85 50
(c) x −4 10 16 24
y 13 23.5 28 34
t (min) 1 2 5 10 15
C (cents) 102 182 422 822 1222
This is an example of interpolation, applying the model (formula) to find C (when P = 100) between the given data points.
This is an example of extrapolation, applying the model to find C (when P = 450) beyond the given data points.
With a linear function:
The gradient m measures the rate of change of the dependent variable y.
The vertical intercept b is the value of the dependent variable y when the independent variable x is zero.
Exercise 1-09:
Linear modelling
x+5 2
--- 1
---(g) Find the cost of a 4-minute phone call using (i) the graph and (ii) the formula. Which is more accurate? Why?
(h) Use the formula to calculate the cost of an 18-minute call. (i) How long did a phone call last if it cost $5.82?
4. The graph shows the value of a computer, depreciating
as a linear relationship.
(a) What is the independent variable? What does it represent?
(b) Find the gradient and vertical intercept. Write the formula for V.
(c) What does the gradient represent?
(d) What does the vertical intercept represent? (e) This linear model does not work for t = 5 and
beyond. Why not?
(f) What was the value of the computer after 2 years? (g) When did the computer have no value (1 decimal place)? (h) When was the computer worth $1585?
5. This table shows the progressive score (total runs) of a cricket team playing in a one-day
50-over match. (An over is a bowler’s round of six balls bowled.) To simplify matters, we assume that it follows a linear model.
(a) Which is the independent variable, and what does it represent?
(b) Which variable would appear on the vertical axis if this linear model were graphed? (c) Graph this relationship.
(d) Find the values of the gradient m and vertical intercept b in the formula S = mn + b. (e) What does the value of m represent?
(f) In cricket, m is also called the run rate. If it is a rate, what units is it measured in? (g) According to this model, what was this team’s final score after 50 overs?
(h) In reality, the graph would tend to flatten at the end, with the gradient becoming smaller. Why?
(i) What was the score after (i) the 2nd over and (ii) the 20th over? (j) After which over was the score (i) 180 runs and (ii) 54 runs?
6. This conversion graph is used to convert distances in miles to kilometres. For example,
25 miles = 40 km. The formula is a linear function of the form K = mM + b. (a) What can you say about the value of the
vertical intercept. Why? (b) What is the dependent variable? (c) Find the gradient of this line. (d) Find a formula for K and use it
to convert:
(i) 100 miles to km (i