Equations PDF
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7
Number and algebra
Equations
Equation-solving has been recorded as far back as 1500 BCE. It was first used in ancient Babylon and Egypt and was brought to Europe from India by the Arabs during the 9th century. The word ‘algebra’ comes from the Arabic word al-jabr , meaning restoration, the process of performing the same operation on both sides of an equation to solve the equation.
NEW CENTURY MATHS ADVANCED for the
A u st ra l i a n C u rr i cu l u m
9
a v o k h s e P / m o c . k c o t s r e t t u h S
n Chapter outline
n Wordbank Proficiency strands
7-01 Equations with variables on both sides 7-02 Equations with brackets 7-03 Equation problems 7-04 Equations with algebraic fractions 7-05 Simple quadratic equations ax 2 c 7-06 Simple cubic equations 3 ax c * 7-07 Equations and formulas 7-08 Changing the subject of a formula* formula*
¼
¼
*STAGE 5.3
cubic equation An equation involving a variable cubed (power of 3), such as 2 x 3 250.
¼
U U U
F F F
R R PS R
U
F
R
formula A rule written as an algebraic equation, using variables.
inverse operation An opposite used in solving an equation, for example, the inverse operation of multiplying is adding
C
U
F
R
C
U U U
F F F
R PS R R
C C C
equation A mathematical statement that two quantities are equal, involving algebraic expressions and an equals sign ( )
linear equation An equation involving a variable that is not raised to a power, such as 2 x 9 17.
þ ¼
quadratic equation An equation involving a variable squared square d (power of 2), such as 3 x 2 6 69.
¼
solution The answer to an equation or problem, the correct value(s) of the variable that makes an equation true solve (an equation) To find the value of an unkno unknown wn variable in an equation
9780170193085
¼
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 Equations
n In this chapter you will: • • • • • •
solve linear linear equations, equations, includin includingg equations equations involving involving simple algebraic algebraic fractions fractions (STAGE 5.3) solve equations involving algebraic fractions solve simple quadra quadratic tic equat equations ions of the the form form ax 2 c (STAGE 5.3) solve simple cubic equations equations of the form ax 3 c use for formul mulas as to to solve solve probl problems ems (STAGE 5.3) change change the subject of a formula
¼
¼
SkillCheck Worksheet StartUp assignment 7 MAT09NAWK10075 Puzzle sheet Solving equations MAT09NAPS00033 Puzzle sheet Backtracking MAT09NAPS00032 Skillsheet Solving equations by balancing MAT09NASS10023
1 Solve each equation. Use substitution to check your solutions. a 3 x
10 ¼ 5 m d 6 ¼ 2 4 2 x ¼ 3 g 5 j w þ 3 ¼ 4 2
b 4
þ 2 y ¼ 21 x 5 e ¼7 4
2a ¼ 17 k a1¼7 6
h 11
c 12 y 5 5r 10 f 3
þ ¼ 23 ¼ i 20 þ 4d ¼ 6 l 6n¼4 3
2 Using n to represent ‘the number’, write an expression for each of these statements. product of the number number and 7. a The product b The square of the number. the sum of the number number and 8. c 5 times the number decreased decreased by 20. d The number product of 6 and the number, number, decreased decreased by nine. e The product number is is even, the next even even number. f If the number
Skillsheet Solving equations by backtrackin backtracking g MAT09NASS10024 Skillsheet
7-01 Equations with with variables variables on both sides sides
Solving equations using diagrams MAT09NASS10025 Puzzle sheet Equations with unknowns on both sides
Summary For equations For equations with variables on both sides, sides , perform operations on both sides to move: • •
all the variabl variables es onto onto one one side side of the equat equation ion all the numbers numbers onto onto the the other other side of the equa equation. tion.
MAT09NAPS00035 Homework sheet Equations 1 MAT09NAHS10012
240
9780170193085
NEW CENTURY MATHS ADVANCED for the
Example
A u st ra l i a n C u rr i cu l u m
9
1
Solve each equations. a 7 x
þ 7 ¼ 2 x þ 2
b 9
6 y ¼ 10 2 y
Solution a
7 x
7 x 2 x 5 x 5 x 7
þ 7 ¼ 2 x þ 2 þ 7 ¼ 2 x 2 x þ 2 þ7¼2 þ 7 ¼ 27 5 x ¼ 5 5 x 5 ¼ 5 5 x ¼ 1
Subtracting 2 x from both sides. Subtracting 7 from both sides. Dividing both sides by 5.
Check: LHS RHS
¼ 7 ð1Þ þ 7 ¼ 0 3
23
1
2
0
¼ RHS ð Þ þ ¼ 9 6 y ¼ 10 2 y 9 6 y þ 2 y ¼ 10 2 y þ 2 y 9 4 y ¼ 10 9 4 y 9 ¼ 10 9 4 y ¼ 1 4 y ¼ 1 4 4 1 y ¼ 4
LHS b
Adding 2 y to both sides. Subtracting 9 from both sides. Dividing both sides by ( 4).
Check: LHS
¼ 9 6 14 ¼ 10 12 1 RHS ¼ 10 2 ¼ 10 1 4 2 LHS ¼ RHS 3
3
Exercise 7-01 Equation Equationss with variable variabless on both sides check k your solutions. 1 Solve each equation, and chec a 5w 3 2w 21 b 2q 10 q 4 c 13 x d 12n 3 5n 11 e 8 y 10 10 y 30 f 3m
þ ¼ þ þ ¼ g 9 2a ¼ a 9 j 12 10u ¼ 20 18u
9780170193085
¼ ¼ h 9 2 x ¼ 18 þ 7 x k 15 7 x ¼ 22 3 x
See Example Example 1
þ 1 ¼ 8 x þ 26 2 ¼ 10 3m i 12 y þ 6 ¼ 6 þ 9 y l 10 6 x ¼ 15 11 x 241
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 Equations
solution A or D 2 For each equation, select the correct solution A,, B B,, C C or D.. a 6 x 1 2 x 11
¼ þ A x ¼ 12 B x ¼ 3 b 11 4 p ¼ 2 p þ 2 A p ¼ 6.5 B p ¼ 2
C x
¼0
C p
¼ 1.5
3 Solve each equation. a 7w 15 w 3 d 50 7 y 20 3 y g 9 t 7t 2
þ ¼ þ b 10 3t ¼ 16 þ t þ ¼ e 8 y 2 ¼ 10 y þ 1 ¼ h 5 y þ 2 ¼ 17 y Select A,, B or D 4 Solve 3n 8 ¼ 7n 12. Select A B,, C C or D.. A n ¼ 5 B n ¼ 2 C n ¼ 1 Just for the record
D x
¼ 2.5
D p
¼3
c 4a f 9 y i 25 D n
þ 2 ¼ 10 4a þ 3 ¼ 9 y 12k ¼ 15 6k
¼ 0.4
Discovering planets
In 1781, British astronomer William Herschel discovered the planet Uranus. At that time, it was the farthest planet known in our solar system. However, astronomers found
y m o n o r t s A n i h c r a e s e R r o f n o i t a i c o s s A a i n r o f i l a C / y r a r b i L o t o h P e c n e S i c
that Uranus’ orbit around the Sun did not follow the expected path. Working separately, mathematicians John Couch Adams of England and Urbain Leverrier of France both predicted that this different orbit was caused by an unknown planet. They calculated the position of this undiscovered planet using a number of equations. In 1846, a German astronomer called Johann Galle located this planet and named it Neptune. The dwarf planet, Pluto, was discovered in a similar manner. How long ago was Pluto discovered?
Worksheet Equations 2 MAT09NAWK10076 Worksheet
7-02 Equations with brackets
Checking solutions MAT09NAWK10078 Puzzle sheet Equations
Summary For equations with brackets (grouping symbols), For equations symbols) , expand the expressions and then solve as usual.
MAT09NAPS00036 Puzzle sheet Equations order activity MAT09NAPS10077
242
9780170193085
NEW CENTURY MATHS ADVANCED for the
Example
A u st ra l i a n C u rr i cu l u m
9
2 Video tutorial
Solve each equation. a 3(a
þ 7) ¼ 6
b 9(m
5) ¼ 7(m þ 1)
3 a 3a
ð þ 7Þ ¼ 6 þ 21 ¼ 6
3a
b
7m
45
7m
MAT09NAVT10022
Expanding the expression to make it a two-step equation. Subtract 21 from both sides.
þ 21 21 ¼ 6 21 3a ¼ 15 3a ¼ 15 3 3 a ¼ 5 9ð m 5Þ ¼ 7ð m þ 1Þ 9m 45 ¼ 7m þ 7
9m
3(2 y 5) ¼ 6(8 3 y)
Can you think of another way to solve this equation without expanding?
Solution a
c 10 y
Equations with brackets
Divide both sides by 3. Check: 3( 5
þ 7) ¼ 3
7m
7
2m 45 ¼ 7 þ 2m 45 þ 45 ¼ 7 þ 45 2m ¼ 52 2m 52 ¼ 2 2 m ¼ 26
3
2
¼6
Expanding brackets on both sides. Subtracting 7m from both sides. Adding 45 to both sides. Dividing both sides by 2.
Check: LHS
¼ 9 ð 26 5Þ ¼ 9 RHS ¼ 7 ð 26 þ 1Þ ¼ 7 LHS ¼ RHS 3
3
3
3
c 10 y 3 2 y 10 y 6 y 4 y 4 y 18 y 22 y
21
¼ 189 27 ¼ 189
ð þ 515Þ ¼¼ 648ð8183 y yÞ þ 15 ¼ 48 18 y þ þ 15 ¼ 48 18 y þ 18 y þ 15 ¼ 48 22 y þ 15 15 ¼ 48 15 22 y ¼ 33 22 y 33 ¼ 22 22 1 y ¼ 1 2
9780170193085
Expanding brackets on both sides. Collecting like terms.
Stage 5.3
Adding 18 y to both sides. Subtracting Subtra cting 15 from both sides. Dividing both sides by 22.
243
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 Equations
Exercise 7-02 Equati Equations ons with brack brackets ets See Example Example 2
1 Solve each equation. a 2(m 3) 8 d 35 7(k 1) g 3h 4(h 6)
þ ¼ ¼ þ ¼ þ j 27 ¼ 7(2 y þ 1)
b 3( x e 4(3 h 6(m
þ 1) ¼ 9 a) ¼ 16 10) ¼ 6 k 5(2 þ 3 p) ¼ 8
which line has has an error error been made in in solving solving 5( x 2 In which 5 x 3 5 x 8
ð Þ ¼ 25 ¼ 25 5 x 8 þ 8 ¼ 25 þ 8 5 x ¼ 33 33 x ¼ 5 ¼ 6 53
Equations with brackets MAT09NAWS10032
Worksheet Word problems with equations
3)
c 5( y f 11 i 8u
2) ¼ 15 y ¼ 9(1 þ 2 p) ¼ 11(u 3) l 22 x ¼ 9(4 x 3)
¼ 25? Select A Select A,, B or D B,, C C or D..
Line 1 Line 2 Line 3 Line 4
A Line 1 B Line 2 Worked solutions
C Line 3
D Line 4
3 Show that k 5 is the solution to 12(k
¼ 1) ¼ 48. 4 Show that a ¼ 6 is the solution to 10 þ a ¼ 2(2 þ a).
5 Solve each equation. a 8(m 2) 5(m 5) b 2( y 3) 4( y 5) d 5( p 2) 3(6 p) e 5n 6 2(2n 1) g 4(3w 1) 5(4 3w) h 2( x 1) 16 5 x
þ ¼ þ þ ¼ þ ¼ þ
¼ þ ¼ þ þ ¼ that the the solution solution to 5(2m 2) ¼ 6(m þ 1) is m ¼ 4. 6 Show that 7 Solve each equation. a 5(m 6) 10 3(m 2) 20 c 7 y 2( y 5) 4( y 10) e 5 y 2( y 3) 4 y 2(2 y 10) g 8 3(1 m) 5(m 3) 4
þ þ þ
þ þ
¼ þ þ ¼ ¼ þþ þþ
c 3(2 x) 4(1 x) f 2(4 3 x) 4(7 3 x) 8 y 5 5(2 y 3) i
þ ¼ þ ¼ ¼
b 3( y 2) 10 2( y 1) 5 d 3 x 4(5 x) 6(2 x) 20 f 11 2(5 y) 4(3 y) 1 h 12 7(2 y 5) 6 15(2 5 y)
þ ¼ þ þ þ ¼ þ þ þ ¼¼ þ
MAT09NAWK10079 Worksheet Angle problems with algebra MAT09MGWK000065 Homework sheet Equations 2 MAT09NAHS10013 Puzzle sheet Writing and solving equations
7-03 Equation problems Word problems can often be solved more easily when they are converted into equations. Follow these steps. • Read the problem problem carefull carefullyy and determine determine what needs needs to be found: ‘What ‘What is the questio question?’ n?’ • Use a variable variable to to represent represent the unknow unknown n quantit quantity. y. • Wri Write te the the probl problem em as as an equ equati ation. on. • Sol ve thethe equati equ ation. on. • Solve Answer Ans wer problem prob lem..
MAT09NAPS00034
244
9780170193085
NEW CENTURY MATHS ADVANCED for the
Example
A u st ra l i a n C u rr i cu l u m
9
3
When three-quarters of a number is decreased by 8 the result is 46. What is the number?
Solution Let the number be x . 3 x 4
8 ¼ 46 3 x 8 þ 8 ¼ 46 þ 8 4 3 x ¼ 54 4 3 x 4 ¼ 54 4 4 3 x ¼ 216 3 x 216 ¼ 3 3 x ¼ 72 3
Translating from words to algebra. Adding 8 to both sides.
Multiplying both sides by 4.
3
Dividing both sides by 3. Check:
The number is 72.
Example
3 3 72 4
8 ¼ 54 8 ¼ 46
4 Technology
A rectangle is three times as long as it is wide. If its perimeter is 60 cm, find its dimensions.
Solution
GeoGebra: Equation problem MAT09MGTC00005
3w cm
Let the width of the rectangle be w cm. Then the length is 3w cm. The perimeter is w 3w w 3w and this is given as 60.
þ þ þ w þ 3w þ w þ 3w ¼ 60 8w ¼ 60 w ¼ 7: 5
w cm
The width of the rectangle is 7.5 cm and the length is 3 3 7.5 22.5 cm. Check: The perimeter of a rectangle with dimensions 7.5 cm and 22.5 cm is 7.5 22.5 7.5 22.5 60 cm.
¼
[
þ
9780170193085
þ
þ
¼
245
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 Equations
Example
5
The sum of three consecutive numbers is 150. Find the numbers.
Consecutive numbers follow each other in order, such as 3, 4, 5.
Solution
Let the first number be x . The next number is x 1 and the third number is x 2. Their sum is x ( x 1) ( x 2) and this is given as 150.
þ
þ
þ þ þ þ x þ x þ 1 þ x þ 2 ¼ 150 3 x þ 3 ¼ 150 3 x ¼ 147 147 x ¼ 3 ¼ 49
The consecutive numbers are 49, 50 and 51. Check: 49 50 51 150. [
þ þ ¼
Example
6
Animated example Applying linear equations MAT09NAAE00008
Justin is 6 years older than his sister Chelsea. Their mother is three times Justin’s age. three ages is 79, write write an a If the sum of the three equation to find Justin’s age. and find each person’s age. b Solve the equation and
Solution a Let x
Justin’s age.
Chelsea’s age is x
¼
6.
Chelsea is 6 years younger than Justin.
The mother’s age is 3 x.
þ ðx 6Þ þ 3 x ¼ 79 5 x 6 ¼ 79 b 5 x 6 ¼ 79 5 x ¼ 85 x ¼ 17 x
a i d e m k a e r b e v a w / m o c . k c o t s r e t t u h S
Justin is 17 years old. Chelseaa is 17 6 11 years old. Chelse Their mother is 3 3 17 51 years old.
¼
Check: 17
246
þ 11 þ 51 ¼¼79
9780170193085
NEW CENTURY MATHS ADVANCED for the
A u st ra l i a n C u rr i cu l u m
9
Exercise 7-03 Equation problems number, the answer is 37. What is the 1 When 7 is subtracted from four times a certain number, number?
See Example Example 3
number, what is the number? number? 2 If 15 more than a number is 3 more than double the number, Two-thirds hirds of a number is 16. What is the number? number? 3 Two-t two-fifths of a numbe numberr is incre increased ased by 15 the resul resultt is 27. What is the number? 4 When two-fifths rectangle is four times as long as it is wide. The perimeter perimeter of the rectangle rectangle is 100 cm. Find 5 A rectangle the dimensions of the rectangle.
See Example Example 4
length of a rectangle rectangle is 7 cm longer than its width. width. 6 The length a Let w be the width of the rectangle. Write an equation for w if the perimeter of the rectangle is 94 cm. the rectangle. b Solve the equation and find the dimensions of the 7 Find the value of x x in this triangle. 88°
2(x + 11)°
the value value of y y . 8 a Find the angle? b What is the size of each alternate angle?
(7 y + 19)°
5( y + 9)°
9 Calculate the size of each marked angle.
Worked solutions Equation problems MAT09NAWS10033
2(4x – 3)° 7(x + 3)°
See
consecutive numbers is 87. Find the numbers. 10 The sum of two consecutive consecutive numbers numbers is 87. Find the numbers. 11 The sum of three consecutive
Example 5
12 The sum of three consecutive even integers is 168. What are the three integers? consecutive odd integers integers is 75. Find the integers. integers. 13 The sum of three consecutive 9780170193085
247
9780170193085
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 Equations
See Example Example 6
years older than his sister, Helen. 14 Dean’s father, Franco, is five times Dean’s age. Dean is eight years The sum of all their ages is 62 years. How old is each person? ( Hint: Let Dean’s age be x .) William iam is three times as old as his daught daughter, er, Rebecca. Rebecca. Rebecca is six years younger younger than her 15 Will brother, Ben. How old is Rebecca if the sum of their three ages is 76 years? total, worth $318. $318. How 16 In my money box I have only $1 and $2 coins. I have 240 coins in total, many $2 coins do I have in the money box? bought four ice creams and received $2.80 $2.80 change from his $10 note. How much did 17 Aerin bought each ice cream cost?
a v e e s s i l E a n e l E / m o c . k c o t s r e t t u h S
twicee as tall as his little sister sister and 30 cm shorter than his father. father. Their combined combined 18 A boy is twic height is 3.8 m. Find (in centimetres) the height of each person. The perimeter of the rectangle is 19 The length of a rectangle is 9.5 cm longer than its width. The 87 cm. Find the dimensions of the rectangle. Janinee is six years younger younger than Paul. Paul is three times times the age of their son Brett. Brett Brett is 20 Janin five years older than his sister Amanda. The sum of all their ages is 125 years. How old is each person?
Worksheet Equations 3
7-04 Equations with algebraic fractions
MAT09NAWS10080 Puzzle sheet
Example
7
Equations code puzzle MAT09NAPS10081 Video tutorial
Solve each equation. x 11 9 3 a 4
þ þ ¼
Solving equations with fractions
Solution
MAT09NAVT00008
a
x
b
þ 11 þ 9 ¼ 3 4
x
þ 11 þ 9 9 ¼ 3 9 4 x þ 11 ¼ 12 4
þ4 11 4 ¼ 12 4 x þ 11 ¼ 48 x þ 11 11 ¼ 48 11 x ¼ 59 x
248
3
3
a
2 ¼ 2a þ 5 4
3
9780170193085
NEW CENTURY MATHS ADVANCED for the
b a
A u st ra l i a n C u rr i cu l u m
9
2 ¼ 2a þ 5 4
3
For equations where all terms are fractions, multiply both sides by a common multiple of the denominators to remove the fractions. The lowest common multiple (LCM) of 3 and 4 is 12, so multiply both sides by 12. 12 3 12
3
3
ða 2Þ 12 ð2a þ 5Þ ða 4 2 Þ ¼¼ 12 4 ð23a þ 5Þ 3
3
4 1
3 1
3 a
ð 2Þ ¼ 4ð2a þ 5Þ 3a 6 ¼ 8a þ 20 5a 6 ¼ 20 5a ¼ 26 26 a¼ 5 1 a ¼ 5 5 Example
8
Solve each equation. a
k
k þ ¼4 3 7
b
4m 5
m 10 ¼4
Solution Multiply both sides sides by 21, the the a Multiply LCM of 3 and 7. k
k
3
7
21 þ ¼ 21 21 3 k 3
7
21 3
þ 21
3 k
3
¼ 84 3 k k ¼ 84 þ 21 3 1 7 1 7k þ 3k ¼ ¼ 84 10k ¼ ¼ 84 k ¼ ¼ 84 10 k ¼ ¼ 8 52 3
7
Multiply ply both sides sides by 10, the b Multi LCM of 5 and 10. 4
4 10 ¼ 10 m
5
m
10
3
4
m 10 3 4 5m 10 3 10 2 10 3 4m 10 1 3 m 5 1 10 1
¼ 40 ¼ 40 8m m ¼ 40 7m ¼ 40 40 m¼ 7 5 m ¼ 5 7
249
9780170193085
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 Equations
Stage 5.3 NSW
Example
9
Solve each equation. 2 x 5 x 7 2 a 3 5
þ ¼
b
Solution
Multiply ply both sides sides by 15, the LCM a Multi of 3 and 5.
2 5 7 þ 15 5 ¼ 15 3 2 5 þ 7 x
5
15
x
3
a
þ 5 þ 3a ¼ 5 6
4
sides by 12, the b Multiply both sides LCM of 6 and 4.
5 3 þ 12 þ 4 ¼ 12 6 þ 5 3 a
2
x
3 15 x 5 ¼ 30 31 1 5ð2 x 5Þ 3ð x þ 7Þ ¼ 30 10 x 25 3 x 21 ¼ 30 7 x 46 ¼ 30 7 x ¼ 76 76 x ¼ 7 ¼ 10 67
2
12 a
61
a
þ 312 4a ¼ 60 1 2ð a þ 5Þ þ 9a ¼ 60 2a þ 10 þ 9a ¼ 60 11a þ 10 ¼ 60 11a ¼ 50 50 a ¼ 11 ¼ 4 116
Exercise 7-04 Equati Equations ons with algebra algebraic ic fracti fractions ons See Example Example 7
1 Solve each equation. a 1 5 15 a 4 y 2 2 6 d 5
þ þ ¼ ¼ g 1 þ 10 x 2 ¼ 10 2 15 d þ 10 ¼ 0 j 7
2 Solve each equation. x 5 x 4 a 4 5 2w 5 3w 1 d 3 5 2 y 4 5 y 2 g 3 5 8 3w 2w 1 j 5 4
þ ¼ þ þ ¼ þ þ ¼
See Example Example 8
250
þ 4 10 ¼ 2 5 pþ1 11 ¼ 8 e b
x
þ 5 þ 9 ¼ 1 3 2w þ 4 ¼ 5 k y
3
8 x
2 4 x
¼ þ 2 4 Select A,, B or D 3 Solve x þ x ¼ 1. Select A B,, C C or D.. 3 2 1 1 A x ¼ 1 B x ¼ 1 C x ¼ 1 5 2
x 1 ¼ 6 6 4 p 10 ¼ 6 l c
2 p
1 ¼ p þ 5
4
3
6 ¼ 2m 3 4 10 12 2 x 9 þ x ¼ 3 i 4 8 x 2 x þ 1 ¼ l
f
3
9
4 ¼ þ 5 3 2 4 x 2 x þ 7 ¼ 4 h 6 7 5 x 1 9 x ¼ k e
i 12
þ 8 ¼ y 2
x
9
4
11
b
c
3
h 3 y
5 1 ¼ 4 5 4 y þ 6 ¼ 15 f 5 þ
m
2
D x
¼2
3
5
3
9780170193085
NEW CENTURY MATHS ADVANCED for the
A u st ra l i a n C u rr i cu l u m
9
4 Solve each equation. d þ ¼ 14 4 3 x x d þ ¼ 10 8 3 4k k ¼ 34 g 3 5
k m m ¼ 3 c þ ¼ 10 2 3 3 2 p p k k ¼ 4 e f ¼ 1 3 5 4 5 3m þ m ¼ 11 4a þ 2a ¼ 10 h i 5 2 5 3 p 1 p þ 3 the solutio solution n to answer A,, B or D 5 What is the 2 ¼ 4? Select the correct answer A B,, C C or D.. 5 A p ¼ 12 B p ¼ 19 C p ¼ 14 D p ¼ 3
a
d
b
k
6 Solve each equation. 2 p p 1 x x 1 0 2 a b 7 4 3 6 2 y 1 y 1 c 2 c 3 4 6 d e 5 2 4 2 7 2 p 1 p 6 y 1 y 2 1 h 8 g 5 2 4 3 3 x 1 5 12u 3 u 4 2 x 2u j k 4 6 7 6
þ ¼ þ þ ¼ þ ¼
þ ¼ ¼ þ ¼ ¼ þ
NSW See Example Example 9
þ 2 þ m þ 1 ¼ 12 2 3 x 1 x 4 f þ 6 ¼ 5 3 a þ 5 2a 3¼ i þ 4 5 m6 þ 2m ¼ 3m 1 l c
þ ¼ þ
Ment Me nta al sk skil ills ls 7A
Stage 5.3
m
5
4
Mat ath hs wi wittho hout ut ca calc lcul ulat ator orss
Fraction of a quantity Learn these commonly-used fractions and their decimal equivalents. 1 2 Decimal 0.5
1 4 0.25
Fraction
1 8 0.125
3 4 0.75
1 5 0. 2
1 10 0.1
1 20 0.05
2 5 0.4
Now we will use them to find a fraction or decimal of a quantity. 1 Study each example. 1 3 72 72 4 4 a 4 18
2 b 3 40 5
¼ ¼
c 3 3 32 4
1
¼ 4 32 ¼8 3 ¼ 24 3
3
e 0:05 3 80
¼
1 3 80 20 4
1
¼ 5 40 ¼8 2 ¼ 16 66 ¼ 1 66 2 ¼ 33 3
3
2
3
3
3
5 3 d 0: 5
3
f 0:125 3 56
¼
1 3 56 8 7
251
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 Equations
2 Now simplify each expression. 1 1 a 3 28 b 3 36 2 4 1 1 3 80 e 3 15 f 5 10 3 1 3 44 3 40 i j 4 8 m 0.1 3 260 n 0.125 3 48 q 0.2 3 70 r 0.5 3 320
1 3 70 10 2 g 3 25 5 c
1 3 64 8 1 3 100 h 20
d
k 0.25 3 60
l 0.4 3 45
o 0.75 3 48 s 0.25 3 56
p 0.05 3 120 t 0.125 3 16
Technology Solv Solving ing equation equationss on a graphics graphics calculator In this activity, you will solve equations on a Casio graphics calculator. Select EQUA (Equation (Equation mode) from the main menu and select Solver select Solver.. 1 Select EQUA equation 2 x 3 15 by pressing 2 X,θ,T + 3 2 At ‘Eq:’, enter the equation ( is SHIFT ) Select SOLV to to solve the equation. 3 Select SOLV
=
15 and EXE .
þ ¼
=
The solution is x 6. The screen also shows LHS (Lft) RHS (Rgt) so the solution is true. Select REPT to to repeat (solve another equation). 4 Select REPT 5 Enter the equation 4(2 x 7) 44 using the ( ) keys. Select SOLV to to solve the equation. What is the solution? 6 Select SOLV x 5 x 1 , we need need to use the the ( ) and a b / c keys. 7 To enter the equation 2 3 Press ( X,θ,T + 5 ) a b / c 2 ( X,θ,T − 1 ) a b / c 3 and EXE .
¼
¼
þ ¼
þ ¼
=
Select SOLV to to solve the equation. What is the solution? 8 Select SOLV equations of your own based on the different different types studied in this chapter. chapter. Use 9 Now write 5 equations your graphics calculator to solve them, and write down the equations and solutions in your book. Swap the equations with other students in your class and try to solve their equations using the graphics calculator.
Investigation: Solving x 2
¼
c
solutions. What are they? 1 x 2 25 has two solutions. solutions for each of the following? following? 2 What are the possible solutions 2 2 2 9 b x 49 100 a x c x 3 What is the inverse operation of ‘squaring’? this example: example: 4 Study this 2 x 81 x 81 which means 81 or 81 x 9 whic wh ich h me mean anss 9 or 9 Check: When x 9, x 2 92 81 When x 9, x 2 ( 9)2 81
¼
¼
¼
¼¼p ffiffi ffi ¼
¼
p ffiffi ffi p ffiffi ffi ¼ ¼ ¼ ¼ ¼ ¼
252
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NEW CENTURY MATHS ADVANCED for the
A u st ra l i a n C u rr i cu l u m
Now use the same method to solve each equation and check your answers. a m 2 1 b k 2 64 each quadratic equation have? have? 5 How many solutions does each 2 2 2 1 b k 64 81 a m c x solutions? (Give reasons for your answers.) 6 Do the following quadratic equations have solutions? 2 2 2 1 b y 64 c h 81 a w that has only one 7 Write an example of a quadratic equation that one solution.
¼
¼
¼
¼
¼
¼
¼
¼
7-05 Simple quadratic equations ax 2
¼
c
An equation in which the highest power of the variable is 2 is called a quadratic equation, equation, for 2 2 2 2 example, x 5, 3 m 7 10, d 4 0 and 4 y 3 y 8.
¼
þ ¼
¼
¼
Summary 2
c (where c is a positive number) has two solutions, x The quadratic equation x (which means x c and x c ).
ffi ¼ p
Example
ffi ¼¼ p
ffic ¼ p
10
Solve each quadratic equation. a y2
¼ 16
b p2
¼ 65
c 5a 2
¼ 245
Solution a y2 y
b p2
¼ 16p
16 ffi ¼¼ 4 ffi ffi
¼ 65p ffiffi ffi p ¼ 65
Finding Findi ng the square root of both sides.
Finding Findi ng the square root of both sides.
65 is not a square number so leave the answer as a surd. c 5a 2
¼ 245 245 a2 ¼ 5 a 2 ¼ 49 p a
49
¼ 7 ffiffi ffi
Dividing both sides by 5.
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 Equations
Example
11
Solve each quadratic equation, writing the solution correct to one decimal place. 3h 2 26 a 4 x 2 600 b 5
¼
¼
Solution a 4 x 2
b 3h 2 5 3h 2
¼ 600 600 x 2 ¼ 4 2 x ¼ 125 p ffiffi ffi ffiffi x ¼ 125 ¼ 11:1803 . . . 11:2
¼ 26 ¼ 26 5 3h 2 ¼ 130 130 h2 ¼ 3 1 h 2 ¼ 43 3 h
3
r ffiffi ffi ffi1ffi
¼ 43 3 5828 . . . ¼ 6: 5828 6: 6
Exercise 7-05 Simple quadratic equations ax 2
¼
See Example Example 10
quadratic equation, writing the solutions solutions as surds if necessary. 1 Solve each quadratic a m 2 144 b x 2 400 c y 2 225
¼ d k ¼ 59 g 8 x ¼ 200 j 5k ¼ 40 k 2 m ¼ 8 2 m2 ¼ 27 p 3 s 3k ¼ 48 2
2
2
2
See Example Example 11
¼ e y ¼ 10 h 9t ¼ 81 k 3w ¼ 30 w2 ¼ 7 n 10 q 8 y ¼ 40 y2 t 2 ¼ 9 5 2
2
2
2
2
¼ f w ¼ 36 a2 ¼ 8 i 2 l 2d ¼ 288 o 4 x ¼ 1 r 2 p þ 3 ¼ 21 u 6 x ¼ 42 2
2
2
2
equation, writing the solution correct to one decimal place. 2 Solve each equation, 2 20 a m b b 2 17 c v2 6
¼ d 2 p ¼ 35 k 2 ¼6 g 16 3w 2 ¼ 20 j 4 2
¼ e 9k ¼ 63 7u 2 ¼ 2 h 10 k a þ 11 ¼ 28 2
2
3 Explain why the quadratic equation k 2
¼ x2 f ¼ 8 5 i 6 y ¼ 84 l 2 y 14 ¼ 63 2
2
25
þ ¼
0 has no solutions. solutions.
quadratic equations has no solutions. Give Give reasons. 4 State which of these quadratic 9 a x2 b 2k 2 5 9 c 3m 2 8 4 9w 2 1 1 d 2 5a 2 3 2 8 d e 4 f 2 3 2
¼ ¼
þ ¼ þ ¼
þ ¼ þ ¼
c
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NEW CENTURY MATHS ADVANCED for the
Investigation: Solving x 3 1 x3
¼
A u st ra l i a n C u rr i cu l u m
9
c
¼ 27 has only one solution. What is it?
to each of the following following equations? 2 What is the solution to 3 3 125 b x 64 8 a x c x3
¼
¼
¼
3 What is the inverse operation of ‘cubing’?
thiss example: 4 Study thi 3 x 27 3 x 27 x 3 The cube root of a negat negative ive numbe numberr is also negat negative. ive. 3 3 Check: When x 3, x ( 3) 27 Now use the same method to solve each equation equation and check your answers. 3 3 1000 b u 216 a r
¼p ffiffi ffi ffi ffi ¼ ¼
¼
¼ ¼
¼
¼
¼
¼
each cubic equation have? have? 5 How many solutions does each 343 c x 3 1 a n3 1 b t 3
¼
3
7-06 Simple cubic equations ax
¼ c
An equation in which the highest power of the variable is 3 is called a cubic equation, equation, for example, 3 3 3 3 2 12, 2m 1 25, d 14 4 and x 3 x 5 x 4 0. x
¼
þ ¼
¼
þ þ ¼
Summary The cubic equation x 3
Example
ffic ¼ c has one solution: x ¼ p 3
12
Solve each cubic equation. a n3
¼ 729
b d 3
¼ 40
c
3
3 y ¼ 1029
Solution a n3
¼ 729 p ffiffi ffi ffiffi n ¼ 729 ¼9 b d 3 ¼ 40 p 40 d ¼ ffiffi ffi 3
3
Finding the cube root of both sides.
Finding Findi ng the cube root of both sides. 40 is not a cube number so leave the answer as a surd.
Stage 5.3 NSW
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 Equations
Stage 5.3
c
3 y 3 ¼ 1029 1029 y 3 ¼ 3 3 y ¼ 343 p ffiffi ffi ffi ffi ffi y ¼ 343
Dividing Dividi ng both sides by 3.
3
7
¼ Example
13
Solve each cubic equation, writing the solution correct to one decimal place. a 2h 3
¼ 18
b
z 3
7
¼ 11
Solution a 2h 3
b z 3 7
¼ 18 18 h3 ¼ 2 3
h
¼ 11 z 3 ¼ 11
9
h
z 3
¼¼ p ffiffi ffi9ffi
3
7
77
ffi ffi ffi ¼ p 77 z ¼ ¼ 4:2543 . . . 4:3
3
3
¼ 2:0800 . . . 2:1
Exercise 7-06 Simple cubic equat equations ions ax 3 See Example Example 12
cubic equation, writing the solutions as surds if necessary. 1 Solve each cubic 27 a r 3 1 b k 3 216 c d 3
¼ ¼ d x ¼ 45 e w ¼ 100 g 4m ¼ 32 h 2t ¼ 250 j 3k ¼ 192 k 7a ¼ 105 n3 ¼ 345:6 m 4q ¼ 665.5 n 5 e3 ¼ 1 q 6 y ¼ 40 p 8 4b 3 12 z 3 s ¼ t þ 4 ¼ 8 5 6 3
¼ f f ¼ 64 3 i c2 ¼ 108 2 p 3 ¼ 4 l 9 o 8 s ¼ 150 r 2v 10 ¼ 1014 u 4 x ¼ 144
3
3
3
3
2
3
3
3
3
See Example Example 13
¼
3
3
equation, writing the solution correct to one decimal place. 2 Solve each equation, 450 a c 3 47 b g 3 151 c y3
¼
3
8 p ¼ 728 g h 3 ¼ 25 20 5v 3 j ¼ 27 8 d
¼ e 3u ¼ 245 h 11a 3 ¼ 80 7 k a 45 ¼ 220 3
3
¼ x f ¼ 11 5 i 6d ¼ 186 l 4 j þ 72 ¼ 166 3
3
3
c
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3
A u st ra l i a n C u rr i cu l u m
cubic equation equation of the form x 3 c always have a solution? a Does a cubic the solution solution to x 3 c positive? b When is the c When does x 3 c have two solutions?
¼
¼
¼
9
Stage 5.3
Worksheet
7-07 Equations and formulas
A formula is an algebraic equation which shows a relationship between variables. For example, the formula is formula for the area of a circle is A pr 2, where A is the area and r is is the radius of the circle (p is a constant). Because the formula is for the area, A is called the subject the subject of of the formula and it is the variable on its own on the left-hand side of the ‘ ’ sign.
¼
¼
Example
14
The formula for the perimeter ( P ) of a rectangle of length l and and width w is given by P 2(l w). Use the formula to find: perimeter of a rectangle rectangle with length 20 cm and width 9 cm a the perimeter
¼ þ
rectangle if its length length is 12 m and its perimeter perimeter is 70 m b the width of a rectangle length of a recta rectangle ngle if its width is 42 cm and its perimeter perimeter is 1.8 m. c the length
Solution
¼ 20, w ¼ 9: P ¼ 2ð l þ wÞ ¼ 2ð20 þ 9Þ ¼ 2 29 ¼ 58
a l
3
[
The perimeter is 58 cm.
¼ 12, P ¼ 70: P ¼ 2ð l þ wÞ 70 ¼ 2ð12 þ wÞ 70 ¼ 24 þ 2w 46 ¼ 2w 46 w¼ 2
b l
[
The23width is 23 m.
¼
(sincee w is given in cm). ¼ 42, P ¼ 1.8 m ¼ 180 cm (sinc P ¼ 2ð l þ wÞ 180 ¼ 2ð l þ 42Þ 180 ¼ 2l þ 84 96 ¼ 2l l ¼ ¼ 962 ¼ 48
c w
[
The length is 48 cm.
Working with formulas MAT09NAWK10082 Homework sheet Equations revision MAT09NAHS10014
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 Equations
Example
15
The cost of hiring a portable sound system for a party is $80 plus $15 per hour. The cost can be repres represented ented by the formula is the total cost C 80 15h where C is
o e d i V d n a o t o h P L l e v a P / m o c . k c o t s r e t t u h S
¼ þ
(in dollars), and h is the number of hours. cost of hiring the the sound system system a Find the cost for 4 hours. willingg to spend $300 for b A family is willin hiring the sound system. What is the maximum number of whole whole rental hours that the family can afford?
Solution
¼ 4: C ¼ ¼ 80 þ 15h
¼ 300: C ¼ ¼ 80 þ 15h
a h
b C
80 15 3 4 140 [ The cost is $140.
300 220
80 15h 15h 220 h 15 2 14 3 [ The maximum number of whole hours is 14.
¼¼ þ
¼¼ þ ¼ ¼
Exercise 7-07 Equati Equations ons and formula formulass 1 Given the formula y 5 x x 5 and b 3 a y if x
¼ þ b, find:
y c b if y y e x if y See Example Example 14
¼ 40 and x¼¼ 3 ¼ 27 and b ¼ 12
x b y if x y d b if y y f x if y
1 and b
16
¼ 6 and x ¼ 1 ¼ 64 and b ¼ 16
temperature in degrees degrees Celsius ( C) can be converted to degrees 2 A temperature 9C Fahrenheit ( F) using the formul formulaa F 32. Convert each 5 temperature to F.
¼ þ ¼
a 35 C
b
10 C
c 16 C
j i k s v o d o v z a R j e g r e S / m o c . k c o t s r e t t u h S
258
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NEW CENTURY MATHS ADVANCED for the
A u st ra l i a n C u rr i cu l u m
9
question 2 to to convert each temperature to C, correct to one decimal place. 3 Use the formula in question 2 a 100 F b 45 F c 78 F 1 a b h is used to find the area of a trapezium, where A is the area, a and b 4 The formula A 2 are the lengths of the parallel sides, and h is the perpendicular height between them. Use the formula to find: trapezium with height height 6 cm and parallel sides sides of length 9 cm and 15 cm a the area of a trapezium
¼ ð þ Þ
height of a trapezium trapezium if its area area is 420 cm 2 and it has parallel sides of length 22 cm and b the height 20 cm length of one side of a trapezium if its parallel parallel side is 20.5 m, its area is 318 m 2 and its c the length height is 12 m. if M M ¼ 12.6 and k ¼ 3.15. Select A Select A,, B or D ¼¼ kS B,, C C or D.. 5
S in in the formula M 5 Find the value of S
A 60
B 7.938
C 20
D 0.8
(in dollars) of hiring a limousine is given by the formula C 6 The cost C (in the number of hours of hire. Find: hiring a limousine limousine for 4 hours hours a the cost of hiring
¼ 180 þ 90h, where h is
See Example Example 15
hiring a limousine limousine for 2 days b the cost of hiring number of hours for which you could could hire a limousine limousine for $720 c the number number of whole hours for which a limousine could could be hired at a cost of $1000. d the maximum number P 5 x 7 The profit, $ P , made by a DVD store is given by P number of DVDs sold. Find: profit made when when 232 DVDs are are sold a the profit number of DVDs sold if the profit profit is $1635. b the number
¼ 900, where x represents the
for a function with P people people using the formula 8 A catering company charges $C for C 75 12.5 P company charge charge for a function with with 10 guests? a How much does the company catering for a group of 60 people. b Find the cost of catering has $640 to spend on catering catering for her next party. What is the maximum maximum number of c Diane has
¼ þ
people she can invite? P 52 and l 4, find w if P P 9 If P A 9
¼
¼
B 12
Select A,, B or D ¼ 2(l þ w). Select A B,, C C or D.. C 18
D 22
(in C) of a hot liquid as it cools is given by the formula T 100 10 The temperature T (in where h is the number of hours it has been cooling. Find: a the temperature of the liquid after 2 hours liquid after 30 minutes b the temperature of the liquid number of hours it takes for the temperature temperature of the liquid liquid to reach 30 C. c the number
¼
17.5h,
cm of a man when 11 Archeologists use the formula H 2.52t 75.8 to estimate the height H cm the tibia (shin) bone length t cm cm is measurable. tibia bone measuring measuring 42 cm long was found. Estimate Estimate the height of the male a An intact male tibia
¼
þ
to the nearest centimetre. length of the tibia bone of a male of height b Estimate, correct to the nearest centimetre, the length 174 cm.
Worked solutions Equations and formulas MAT09NAWS10034
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 Equations
Stage 5.3
7-08 Changing the subject of a formula
NSW Puzzle sheet Formulas squaresaw MAT09NAPS10083
In the formula v u at , v is the subject the subject of of the formula. When the formula is rearranged so that one of the other variables becomes the subject, the process is called changing the subject of the formula.. To change the subject of a formula, use the same rules as for solving an equation. The formula
¼ þ
answer is not a number but an algebraic expression (another formula).
Example
16
the formula formula v a For the
to a . ¼ u þ at , change thep subject ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 2 b Make b the subject of the formula x ¼ b 4ac. m1 , change the subject of the formula to m . c Given P ¼ mþ1
Solution a To make a the subject of the formula, solve it like an a
equation for .
¼ u þ at u þ at ¼ v at ¼ v u vu a¼ t p ffiffi ffi ffi ffi ffi ffi ffi ffi ffi x ¼ b 2 4ac p ffiffi 2ffi ffi ffi ffi ffi ffi ffi ffi b 4ac ¼ x b 2 4ac ¼ x 2 b 2 ¼ x 2 þ 4ac v
b
b
c
x 2
4 p ffi ffi ffi ffi ffi ffi ¼ þ ffi ffi ffiffi
¼ mm þ 11 m1 ¼ P mþ1 m 1 ¼ Pðm þ 1Þ ¼ Pm þ P m Pm ¼ P þ 1 mð1 P Þ ¼ P þ 1 P þ 1 m¼ 1 P
P
ac
Swapping sides so new subject a appears on the LHS. Subtracting u from both sides. Dividing Dividi ng both sides by t t .
Swapping sides. Squaring both sides. Adding 4ac to both sides. Taking the positive and negative square root of both sides.
Swapping sides. Multiplying both sides by m m Expanding.
þ 1.
Moving the m -terms to the LHS, the 1 to the RHS. Factorise m from the LHS. Dividing both sides by 1
P .
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A u st ra l i a n C u rr i cu l u m
Exercise 7-08 Changi Changing ng the the subjec subjectt of of the the formula formula the subject of each formula formula to x . 1 Change the a d x c b y mx b
¼ þ x d y ¼ þ p m x
g A
stage 5.3 See Example Example 16
¼ þ e k ¼ mrx
9
c p
i A
¼ ax y m f v ¼ x
y
¼ þ2
h c
¼ ax þ by
¼ 12 hð x þ yÞ
1 volume of a pyramid has the formula V Ah, where A is the area of the base and h is 2 The volume 3 the perpendicular height. Which of the following is the correct formula for A ? Select the correct answer A answer A,, B or D B,, C C or D.. 1 3 3V Vh A A B A C A 3Vh D A 3 Vh h 3 Make y the subject of each formula. x a m an ay b x y 2 c k
¼
¼ þ ffi p d Q ¼ P þ y r g m ¼ p ffi yffi
¼ ¼
¼
¼
¼ e x þ y ¼ r h b ¼ c 2ay 2
2
¼¼ y f M ¼ ny yffi ffiffi ffi xffi ffi i t ¼ ¼ p
2
2
2
¼
4 Change the subject of each formula to the pronumeral shown. 2
¼ r r ¼ ? c K ¼ ¼ 12 mv 2 v ¼ ? 1 h e s ¼ ut þ at 2 a ¼ ? f A ¼ ð x þ yÞ y ¼ ? 2 2 5 What is the correct formula for p ¼ m(n þ x) if x x is the subject? Select the correct answer ¼ lbh h ¼ ? ¼ 9 5C þ 32 C ¼ ? d F ¼ a V
b A
p
or D A, B B,, C C or D.. A x
¼ p mn
B x
p ¼ mn
¼ mp n
C x
D x
¼ pn m
6 Solve each equation for a . a
a
¼ ap þ q d p ¼ ax a þ y
5
¼ 3(a þ b) e k ¼ ¼ 11 þ aa b ar
¼ D 2a f M (a þ b) ¼ N (a b) c Da
is the distance travelled in 7 The cost ($ C ) of a hire car is given by C C 80 4.2d , where d is kilometres. cost of hiring hiring the car for a journey journey of 50 km. a Find the cost
¼ þ
the subject of the formula. b Make d the kilometress that can be travel travelled led in the hire car for $402. c Find the number of whole kilometre angle sum of a shape with n sides is A , where A 180(n 2). 8 The angle formula to find the angle sum of a shape with 7 sides. a Use the formula b Make n the subject of the formula. angle sum of a polygon polygon is is 1440 , how many sides does it have? c If the angle the subject of the formula 1 1 1. 9 Make r the
¼
x
¼ r þ s
m , where m is the adult’s mass h2
body-mass index of an adult is given by the formula B 10 The body-mass
in kilograms and h is their height in metres. Changee the subject of the the formula to h . a Chang Hencee find, correct to two decimal places, places, the height of a person with a body mass index of b Henc 25 and a mass of 60 kg.
¼
Worked solutions Changing the subject of a formula MAT09NAWS10035
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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 Equations
Me Men nta tall ski kill llss 7B
Math Ma thss wi with thou outt ca calc lcul ulat ator orss
Percentage of a quantity Learn these commonly-used percentages and their fraction equivalents. 50 0% Percentage 5 1 Fraction 2
25% 1 4
12. 5% 1 8
75% 3 4
20% 1 5
1 33 3 % 1 3
10% 1 10
2 66 3 % 2 3
Now we will use them to find a percentage of a quantity. 1 Study each example. 1 3 25 a 20% 3 25 5 5
¼ ¼
5% 3 32 c 12: 5%
¼ 18
3
32
¼ 12 120 ¼ 60 3 56 ¼ 60 4
b 50% 3 120
d 75% 3
4
¼ 13 ¼9
3
1 60 ¼ 4 3
¼ 1 e 33 % 3 27 3
3
3
¼ 15 3 ¼ 45 60 ¼ 2 60 3 1 ¼ 3 60 ¼ 20 2 ¼ 40
3
3
3
27
2 f 66 % 3 3
3
3
3
2
3
2 Now simplify each expression. 1 a 25% 3 44 b 33 % 3 120 3 1 e 10% 3 230 f 12 % 3 48 2 1 i 75% 3 24 j 33 % 3 45 3 2 m 12.5% 3 88 n 66 % 3 21 3
c 20% 3 35
g 50% 3 86
k 25% 3 160 o 20% 3 60
2 d 66 % 3 36 3 h 20% 3 400
l 10% 3 650 p 75% 3 180
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A u st ra l i a n C u rr i cu l u m
Power plus 1 Solve each equation. 5 14 a 2 y
¼
b
2
7
1 ¼ r r 3 2
12 8 4 2 3 3 m 5 2 m 1 e 4 3
2
þ ¼ d 4m 3 ¼ m þ 72 ð þ Þ ð Þ ¼ m 10 þ m 6 2 Y , find: 2 Given that W ¼ ¼ X X þ þ Y when X ¼ 15 and Y ¼ 10 b W when when X ¼ 6 and Y ¼ 12 a W when when W ¼ 25 and Y ¼ 6 d Y when when W ¼ 5 and X ¼ 1 c X when c y
raises pigs and chickens. chickens. From a total of 42 animals animals she can count 116 legs. 3 A farmer raises Write an equation and solve it to find how many chickens she has. old as his daughter. daughter. Ten years years ago he was three times times as old as her. 4 A man is twice as old Write an equation and solve it to determine how old his daughter is now. 5 Consider x 2 y 2 4. value for x is 2 and the largest value for x is 2. a Explain why the smallest value restrictions on the values that y can take? Explain why. b Are there any restrictions 4 x 2 . c By making y the subject, show that y
þ ¼
p ffiffi ffi ffi ffi ffi ffi ¼
9
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Chapter 7 review n Language of maths Puzzle sheet Equations crossword MAT09NAPS10084
algebraic fraction
brackets
consecutive
cube root
cubic equation
equation
expand
formula
inverse operation
LHS
lowest common multiple (LCM)
linear equation
quadratic equation
RHS
solution’
solve
square root
subject
substitute
surd
undoing
unknown
variable
equations involves using inverse operations operations on both sides of the 1 Which method of solving equations equation? the subject of the formula A 2 What is the subject of
¼ 12 ða þ bÞh
given to numbers that follow follow each other other in order, such as 9, 10, 11? 3 What name is given 4 Write an example of: quadratic equation equation a a quadratic
linear equation. b a linear
linear equation have? have? 5 How many solutions does a linear does LHS stand stand for? 6 What does LHS
n Topic overview Worksheet Mind map: Equations (Advanced) MAT09NAWK10086
Copy (or print) and complete this mind map of the topic, adding detail to its branches and using pictures, symbols and colour where needed. Ask your teacher to check your work. Equations with variables on both sides
Changing the subject of a formula
Equations and formulas
Equations with brackets
Equations
Simple quadratic and cubic equations
Equation problems
Equations with algebraic fractions
264
9780170193085
Chapter 7 revision 1 Solve each equation. Check your solutions. a 2 x 5 3 x 4 b 7 4 x x 8 d 3 x 4 2 x 7 e 5n 3 2n 15 g 4t 12 4t h 8 j 17 10 j
See Exercise Exercise 7-01
¼ 3w 7 ¼ 5d 71 ¼8q 2 Write an equation with x on both sides that has the solution x ¼ 3. þ ¼ þ þ ¼ þ ¼
c 5 6w f 2d 8 i 6 3q
þ ¼ ¼ ¼
3 Solve each equation. a 2(w 5) 4 b 3(1 d 2(3 x) 5( x 1) e 3(1
¼ þ ¼ þ
þ 4n) ¼ 15 y) ¼ 4(2 y)
c 5(1 f 2(3
See Exercise Exercise 7-01 See Exercise Exercise 7-02
3 p) ¼ 20 4 x) ¼ (2 x þ 3)
length of a rectangle rectangle is 6 cm longer than it is wide. The perimeter perimeter of the rectangle is 4 The length 76 cm. Find the dimensions of the rectangle.
See Exercise Exercise 7-03
value of each pronumeral. 5 Find the value
See Exercise Exercise 7-03
a
b (2a + 15)° 4x° 107°
56°
number, what is the number? number? 6 If 6 more than a number is the same as 5 more than double the number,
See Exercise Exercise 7-03
7 Solve each equation. 8n 6 d 4 9 4 a b 2 3 8 Solve each equation. x 1 x 4 7n 3 6 b a 5 2 5 equation. 9 Solve each quadratic equation.
See Exercise Exercise 7-04
þ ¼ 6n þ 5 2
þ ¼
3
c
2 y
1 þ y þ 1 ¼ 6
4
c 3 z 2
¼
¼
p2 ¼ 4 2
¼
c
¼ 105
t 3
2
¼ 62: 5 5
mass index (BMI) (BMI) of an adult is B M 2 , where M is is the mass in kilograms and 11 The body mass h h is the height in metres. whole number the BMI BMI of Dean who is 1.85 m tall tall and has a mass of 72 kg. a Find as a whole mass of a person with a BMI of 24, who who is 2.1 m tall. b Find the mass
¼
is the distance travelled in 12 The cost, C , in dollars, of hiring a taxi is C 5 2.4d , where d is kilometres. Find: distance travelled travelled is 15 km a the cost of a taxi trip if the distance b the distance travelled if the cost of a taxi trip was $78.20.
¼ þ ¼
13 Make w the subject of each formula. a a
¼ kw þ v
b p
¼ mðt wÞ
See Exercise Exercise 7-04 See Exercise Exercise 7-05
a d 2 64 b 8 p 2 288 equation. 10 Solve each cubic equation. a x 3 1331 b 4h 3 864
¼
2 p 3
Stage 5.3
þ 5n þ 4 ¼ 2
þ ¼
c
c x
1
¼ w þ y
Stage 5.3 See Exercise Exercise 7-06 See Exercise Exercise 7-07
See Exercise Exercise 7-07
Stage 5.3 See Exercise Exercise 7-08
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