Cambridge Maths 7 Chapter 9

(1)

Chapter

9

Equations 1

What you will learn

Introduction to equations

Solving equations by inspection

Equivalent equations

Solving equations systematically

Equations with fractions

Equations with brackets

Formulas and relationships

Formulas and relationships EXTENSIONEXTENSION

Using equations to solve problems

Using equations to solve problems EXTENSIONEXTENSION

9A 9A 9B 9B 9C 9C 9D 9D 9E 9E 9F 9F 9G 9G 9H 9H

(2)

Theme park equations

Equations are used widely in mathematics and in

many other ﬁelds. Whenever two things are

many other ﬁelds. Whenever two things are equal, orequal, or

should be equal, there is the potential to use the study

of equations to help deal with such a situation.

Knowledge of mathematics and physics is

vitally important when designing theme park rides.

Engineers use equations to ‘build’ model rides on a

computer so that safety limits can be determined in a

virtual reality in which no one gets injured.

Algebraic equations are solved to determ

Algebraic equations are solved to determine theine the

dimensions and strengths of structures

dimensions and strengths of structures required torequired to

deal safely with the

deal safely with the combined forces of weight, speedcombined forces of weight, speed

and varying movement. Passengers might scream

with a mixture

with a mixture of terror and of terror and excitemexcitement but they mustent but they must

return unharmed to earth!

At Dreamworld on the Gold Coast, Queensland,

‘The Claw’ swings 32 people upwards at 75 km/h to a

maximum height of 27

maximum height of 27..1 m (1 m (9 storeys)9 storeys), simultaneously, simultaneously

spinning 360° at 5 r.p.m

spinning 360° at 5 r.p.m. (revolutions per . (revolutions per minute).minute).

‘The Claw’ is the most powerful pendulum ride on the

planet. It is built to scare!

NSW Syllabus

for the Australian

Curriculum

Strand: Number and Algebra

Substrand: EQUATIONS

Outcome

A student uses algebraic techn

A student uses algebraic techniquesiques

to solve simple linear and

to solve simple linear and quadraticquadratic

equations.

(MA4–10NA)

(3)

1

1 Fill in the missing number.Fill in the missing number.

a

a ++₃₃==₁₀₁₀ _b_b ₄₁₄₁−− ==₂₁₂₁ _c_c ××₃₃==₄₈₄₈ _d_d ₁₀₀₁₀₀÷÷ ==₂₀₂₀

2

2 IfIf ==_5,_5,_{state whether each of these equations is true or}_{state whether each of these equations is true or}_false._false.

a

a −−₂₂==₅₅ _b_b ××₃₃==₁₅₁₅ _c_c ₂₀₂₀÷÷₄₄== _d_d ×× ==₃₆₃₆

3

3 IfIf x x ==_{3, ﬁnd the value of:}_{3, ﬁnd the value of:}

a

a x x ++₄₄ _b_b ₈₈−−_x_x _c_c ₅₅_x_x _d_d ₃₃++₇₇_x_x

4

4 IfIfnn ==_{6, state the value of:}_{6, state the value of:}

a

a nn÷÷₂₂ _b_b ₄₄_n_n ++₃₃ _c_c ₈₈−−_n_n _d_d ₁₂₁₂÷÷_n_n ++₄₄

5

5 The expressionThe expressionnn ++_{3 can be described as ‘the sum of}_{3 can be described as ‘the sum of} _n_n_{and 3’. Write expressions for:}_{and 3’. Write expressions for:}

a

a the sum ofthe sum of k k and 5 and 5 bb doubledouble p p

c

c the product of 7 andthe product of 7 and y y dd one-half ofone-half of qq

6

6 Simplify each of the following algebraic expressions.Simplify each of the following algebraic expressions.

a

a 33 x x ++₂₂_x_x _b_b ₇₇××₄₄_b_b _c_c ₂₂_a_a ++₇₇_b_b++₃₃_a_a _d_d ₄₄++₁₂₁₂_a_a −−₂₂_a_a ++₃₃

7

7 State the missing values in the tables below.State the missing values in the tables below.

a a n n 11 33 cc d d 1212 5 5×× n n aa bb 2200 3355 ee b b n n 22 44 cc 1212 ee n n−− 22 aa bb 1010 d d 3939 c c n n aa bb 33 00 ee 2 2 n n++11 55 1111 cc d d 2727 d d n n aa 44 cc d d 22 6 6−− n n 33 bb 55 66 ee 8

8 For each of the following, state the opposite operation.For each of the following, state the opposite operation.

a a ×× _b_b ++ _c_c ÷÷ _d_d −−

P

r

e

-

- t

t

e

s

t

(4)

Introduction to

equation

s

An equation is a mathematical statement used to

say that two expressions have the same value. It

will always consist of two expressions that are

separated by an equals sign (

separated by an equals sign (==_)._).

Sample equations include:

3 3++₃₃==₆₆ 30 30==₂₂××₁₅₁₅ 100 100−−₃₀₃₀==₆₀₆₀++₁₀₁₀

which are all true equations.

An equation does not have to be true.

For instance: For instance: 2 2++₂₂==₁₇₁₇ 5 5==₃₃−−₁₁ _and_and 10

10++₁₅₁₅==₁₂₁₂++₃₃ _are_are_all_all_false_false_equations._equations.

If an equation contains pronumerals, one cannot tell whether the equation is true

If an equation contains pronumerals, one cannot tell whether the equation is true or false until valuesor false until values

are substituted for the

are substituted for the pronumerals.pronumerals.

For example, 5

For example, 5 ++_x_x==_{7 could be true (if}_{7 could be true (if}_x_x_{is 2) or}_{is 2) or}_{it could be false (if}_{it could be false (if}_x_x_{is 15).}_{is 15).}

Let’s start:

Equations – T

rue or

false?

Rearrange the following ﬁve symbols to make as many different equations as possible.

5, 2, 3,

5, 2, 3, ++_,_, ==

•

• Which of them are true? Which are false?Which of them are true? Which are false?

•

• Is it always possible to rearrange numbers and operations to make true Is it always possible to rearrange numbers and operations to make true equations?equations?

9A

This equation was proposed by

This equation was proposed by the famous scientist Albertthe famous scientist Albert

Einstein (1879–1955). It explains the

Einstein (1879–1955). It explains the special theory of relativity.special theory of relativity.

■

■ AnAnexpressionexpression is a collection of pronumerals, numbers and operators without an equals signis a collection of pronumerals, numbers and operators without an equals sign (e.g. 2

(e.g. 2 x x ++_3)._3).

■

■ AnAnequationequation is a is a mathematimathematical statement stating that two things are equal (e.g. cal statement stating that two things are equal (e.g. 22 x x ++ 3 3== 4 4 y y−− 2). 2).

■

■ Equations have a left-hand side (LHS), a right-hand side (RHS) and Equations have a left-hand side (LHS), a right-hand side (RHS) and an equals sign in an equals sign in between.between. 2 2 x x ++₃₃ LHS LHS = = ₄₄_y_y−−₂₂ RHS RHS equals sign equals sign ■

■ Equations are mathematical statements that can be true (e.g. 2Equations are mathematical statements that can be true (e.g. 2 ++ 3 3== 5) or false 5) or false (e.g. 5

(e.g. 5++₇₇==_21)._21).

■

■ If a pronumeral is If a pronumeral is included in an equation, you need included in an equation, you need to know the value to substitute beforeto know the value to substitute before deciding whether the equation is true.

deciding whether the equation is true.

For example, 3

For example, 3 x x ==_{12 would be true if 4 is substituted for}_{12 would be true if 4 is substituted for}_x_x_{, but it would be false if 10 is substituted.}_{, but it would be false if 10 is substituted.}

■

■ The value(s) that make an equation true are calledThe value(s) that make an equation true are called solutionssolutions.. For example, the solution of 3

For example, the solution of 3 x x ==_{12 is}_{12 is}_x_x==_4._4.

K K e e y y i i d d e e

a a s s

(5)

Example 1

Identifying equations

Which of the following are equations?

a

a 33++₅₅==₈₈ _b_b ₇₇++₇₇==₁₈₁₈ _c_c ₂₂++₁₂₁₂ _d_d ₄₄==₁₂₁₂−−_x_x _e_e ₃₃++_u_u

S

SOOLLUUTTIIOONN EEXXPPLLAANNAATTIIOONN

a

a 33++₅₅==₈₈_i_i_s_s_a_a_n_n_e_e_q_q_u_u_a_a_t_t_i_i_o_o_n_n_._. _T_T_h_h_e_e_r_r_e_e_a_a_r_r_e_e_t_t_w_w_o_o_e_e_x_x_p_p_r_r_e_e_s_s_s_s_i_i_o_o_n_n_s_s₍₍_i_i_._._e_e_._.₃₃++_{5 and 8)}_{5 and 8)}_separated_separated

by an equals sign.

b

b 77++₇₇==₁₁₈₈_i_i_s_s_a_a_n_n_e_e_q_q_u_u_a_a_t_t_i_i_o_o_n_n_._. _T_T_h_h_e_e_r_r_e_e_a_a_r_r_e_e_t_t_w_w_o_o_e_e_x_x_p_p_r_r_e_e_s_s_s_s_i_i_o_o_n_n_s_s_s_s_e_e_p_p_a_a_r_r_a_a_t_t_e_e_d_d_b_b_y_y_a_a_n_n_e_e_q_q_u_u_a_a_l_l_s_s_s_s_i_i_g_g_n_n_._.

Although this equation is false, it is still an

Although this equation is false, it is still an equation.equation.

c c 22++₁₁₂₂_i_i_s_s_n_n_o_o_t_t_a_a_n_n_e_e_q_q_u_u_a_a_t_t_i_i_o_o_n_n_._. _T_T_h_h_i_i_s_s_i_i_s_s_j_j_u_u_s_s_t_t_a_a_s_s_i_i_n_n_g_g_l_l_e_e_e_e_x_x_p_p_r_r_e_e_s_s_s_s_i_i_o_o_n_n_._._T_T_h_h_e_e_r_r_e_e_i_i_s_s_n_n_o_o_e_e_q_q_u_u_a_a_l_l_s_s_s_s_i_i_g_g_n_n_._. d d 44==₁₂₁₂ −−_x_x_i_i_s_s_a_a_n_n_e_e_q_q_u_u_a_a_t_t_i_i_o_o_n_n_._. _T_T_h_h_e_e_r_r_e_e_a_a_r_r_e_e_t_t_w_w_o_o_e_e_x_x_p_p_r_r_e_e_s_s_s_s_i_i_o_o_n_n_s_s_s_s_e_e_p_p_a_a_r_r_a_a_t_t_e_e_d_d_b_b_y_y_a_a_n_n_e_e_q_q_u_u_a_a_l_l_s_s_s_s_i_i_g_g_n_n_._. e e 33++_u_u_i_i_s_s_n_n_o_o_t_t_a_a_n_n_e_e_q_q_u_u_a_a_t_t_i_i_o_o_n_n_._. _T_T_h_h_e_e_r_r_e_e_i_i_s_s_n_n_o_o_e_e_q_q_u_u_a_a_l_l_s_s_s_s_i_i_g_g_n_n_,_,_s_s_o_o_t_t_h_h_i_i_s_s_i_i_s_s_n_n_o_o_t_t_a_a_n_n_e_e_q_q_u_u_a_a_t_t_i_i_o_o_n_n_._.

Example 2

Classifying equations

For each of the following equations, state whether it is true or false.

a

a 77++₅₅==₁₂₁₂ _b_b ₅₅++₃₃==₂₂××₄₄

c

c 1212××₍₂₍₂−−₁₎₁₎==₁₄₁₄ ++₅₅ _d_d ₃₃++₉₉_x_x==₆₀₆₀ ++_{6, if}_{6, if}_x_x==₇₇

e

e 1010++_b_b ==₃₃_b_b++_{1, if}_{1, if}_b_b ==₄₄ _f_f ₃₃++₂₂_x_x==₂₁₂₁ −−_y_y_{, if}_{, if}_x_x==_{5 and}_{5 and}_y_y==₈₈

S

a

a ttrruuee TThhe e lleefftt--hhaannd d ssiidde e ((LLHHSS) ) aannd d rriigghhtt--hhaannd d ssiidde e ((RRHHSS))

are both equal to 12, so the equation is true.

b

b ttrruuee LLHHSS==₅₅++₃₃==_{8 and RHS}_{8 and RHS}==₂₂××₄₄==_{8, so both sides}_{8, so both sides}

are equal.

c

c ffaallssee LLHHSS==_{12 and RHS}_{12 and RHS}==_{19, so the equation is false.}_{19, so the equation is false.}

d

d ttrruuee IIff x x is 7, then: is 7, then:

LHS

LHS ==₃₃++₉₉××₇₇==₆₆₆₆

RHS

RHS==₆₀₆₀++₆₆==₆₆₆₆

e

e ffaallssee IIffbb is 4, then: is 4, then:

LHS

LHS ==₁₀₁₀ ++₄₄==₁₄₁₄

RHS

RHS==₃₍₄₎₃₍₄₎++₁₁==₁₃₁₃

f

f truetrue IfIf x x ==_{5 and}_{5 and}_y_y==_{8, then:}_{8, then:}

LHS

LHS ==₃₃++₂₍₅₎₂₍₅₎==₁₃₁₃

RHS

(6)

Example 3

Writing equations from a description

Write equations for each of the following scenarios.

a

a The sum ofThe sum of x x and 5 is 22. and 5 is 22.

b

b The number of cards in a deck isThe number of cards in a deck is x x . In 7 decks there are 91 . In 7 decks there are 91 cards.cards.

c

c Priya’s age is currentlyPriya’s age is currently j j. In 5 years’ time her age will equal 17.. In 5 years’ time her age will equal 17.

d

d Corey earns $Corey earns $ww per year. He spends per year. He spends 11 12

12 on spoon sport rt andand 2 2 13

13 on on food. The total amount Corey spendsfood. The total amount Corey spends

on sport and

on sport and food is $15 000.food is $15 000.

S

a

a x x ++₅₅==₂₂₂₂ _T_T_h_h_e_e _s_s_u_u_m_m _o_o_f_f_x_x_{and 5 is written}_{and 5 is written}_x_x++_5._5.

b

b 77 x x ==₉₁₉₁ ₇₇_x_x_means_means₇₇××_x_x_{and this number must equal the 91 cards.}_{and this number must equal the 91 cards.}

c

c j j++₅₅==₁₁₇₇ _I_I_n_n₅₅_y_y_e_e_a_a_r_r_s_s_’_’_t_t_i_i_m_m_e_e_P_P_r_r_i_i_y_y_a_a_’_’_s_s_a_a_g_g_e_e_w_w_i_i_l_l_l_l_b_b_e_e₅₅_m_m_o_o_r_r_e_e_t_t_h_h_a_a_n_n_h_h_e_e_r_r_c_c_u_u_r_r_r_r_e_e_n_n_t_t

age, so

age, so j j++_{5 must be 17.}_{5 must be 17.}

d d 11 12 12 2 2 13 13 15000 15000 × × w w + + × × ww== 11 12

12 of Corey’of Corey’s wage s wage isis 1 1 12 12 ×× w w and and 2 2 13

13 of his of his wage iswage is 2 2 13 13×× w w.. 1

1 Classify each of the following as an equation (E) Classify each of the following as an equation (E) or not an or not an equation (N).equation (N).

a a 77++_x_x==₉₉ _b_b ₂₂++₂₂ _c_c ₂₂××₅₅==_t_t d d 1010==₅₅++_x_x _e_e ₂₂==₂₂ _f_f ₇₇××_u_u g g 1010÷÷₄₄==₃₃_p_p _h_h ₃₃==_e_e++₂₂ _i_i _x_x++₅₅ 2

2 Classify each of these equations as true or Classify each of these equations as true or false.false.

a

a 22++₃₃==₅₅ _b_b ₃₃++₂₂==₆₆ _c_c ₅₅−−₁₁==₆₆

3

3 Consider the equation 4Consider the equation 4++₃₃_x_x==₂₂_x_x++_9._9.

a

a IfIf x x ==_{5, state the value of the left-hand side (LHS).}_{5, state the value of the left-hand side (LHS).}

b

b IfIf x x ==_5,_5,_{state the value of the right-hand side (RHS).}_{state the value of the right-hand side (RHS).}

c

c Is the equation 4Is the equation 4 ++₃₃_x_x==₂₂_x_x++_{9 true or false when}_{9 true or false when}_x_x==_5?_5?

4

4 IfIf x x ==_{2, is 10}_{2, is 10}++_x_x==_{12 true or}_{12 true or}_false?_false? Example 1 Example 1 Example 2a–c Example 2a–c Example 2d,e Example 2d,e

Exercise 9A

W

W O O R RK K I I N N G G

M M A A T T H H _E_E M

M _A_A_T_T I I C C A A L L

L L Y Y U U FF R R PSPS C C

(7)

5

5 For each of the following equations, state whether it is true or false.For each of the following equations, state whether it is true or false.

a a 1010××₂₂==₂₀₂₀ _b_b ₁₂₁₂ ××₁₁₁₁==₁₄₄₁₄₄ _c_c ₃₃××₂₂==₅₅++₁₁ d d 100100−−₉₀₉₀==₂₂××₅₅ _e_e ₃₀₃₀ ××₂₂==₃₂₃₂ _f_f ₁₂₁₂−−₄₄==₄₄ g g 2(32(3−−₁₎₁₎==₄₄ _h_h ₅₅−−₍₂₍₂++₁₎₁₎==₇₇−−₄₄ _i_i ₃₃==₃₃ j j 22==₁₇₁₇−−₁₄₁₄−−₁₁ _k_k ₁₀₁₀ ++₂₂==₁₂₁₂ −−₄₄ _l_l ₁₁××₂₂××₃₃==₁₁++₂₂++₃₃ m m 22××₃₃××₄₄==₂₂++₃₃++₄₄ _n_n ₁₀₀₁₀₀−−₅₅××₅₅==₂₀₂₀××₅₅ _o_o ₃₃−−₁₁==₂₂++₅₅−−₅₅ 6

6 IfIf x x ==_{3, state whether each of these equations is true or false.}_{3, state whether each of these equations is true or false.}

a

a 55++_x_x==₇₇ _b_b _x_x++₁₁==₄₄ _c_c ₁₃₁₃ −−_x_x==₁₀₁₀ ++_x_x _d_d ₆₆==₂₂_x_x

7

7 IfIfbb ==_4,_4,_{state whether each of the following equations is true or false.}_{state whether each of the following equations is true or false.}

a

a 55bb ++₂₂==₂₂₂₂ _b_b ₁₀₁₀ ××₍₍_b_b −−₃₎₃₎==_b_b ++_b_b ++₂₂

c

c 1212−−₃₃_b_b==₅₅−−_b_b _d_d _b_b ××₍₍_b_b++₁₎₁₎==₂₀₂₀

8

8 IfIfaa ==_{10 and}_{10 and}_b_b ==_{7, state whether each of these equations is true or false.}_{7, state whether each of these equations is true or false.}

a a aa++_b_b==₁₇₁₇ _b_b _a_a ××_b_b==₃₃ _c_c _a_a××₍₍_a_a −−_b_b₎₎==₃₀₃₀ d d bb××_b_b==₅₉₅₉−−_a_a _e_e ₃₃_a_a==₅₅_b_b −−₅₅ _f_f _b_b××₍₍_a_a −−_b_b₎₎==₂₀₂₀ g g 2121−−_a_a ==_b_b _h_h ₁₀₁₀−−_a_a ==₇₇−−_b_b _i_i ₁₁++_a_a −−_b_b ==₂₂_b_b −−_a_a 9

9 Write equations for each of the following.Write equations for each of the following.

a

a The sum of 3 andThe sum of 3 and x x is equal to 10. is equal to 10.

b

b WhenWhenk k is multiplied by 5, the result is 1005. is multiplied by 5, the result is 1005.

c

c The sum ofThe sum ofaa and and bb is 22. is 22.

d

d WhenWhend d is doubled, the result is 78. is doubled, the result is 78.

e

e The product of 8 andThe product of 8 and x x is 56. is 56.

f

f WhenWhen p p is tripled, the result is 21. is tripled, the result is 21.

g

g One-quarter ofOne-quarter oft t is 12. is 12.

h

h The sum ofThe sum ofqq and and p p is equal to the product of is equal to the product of qq andand p p.. Example 2f Example 2f Example 3a Example 3a W W M M A A T T H H _E_E M

M _A_A_T_T I I C C A A

L L L L Y Y U U FF R R PSPS C C 10

10 Write true equations for each of these problems. You do notWrite true equations for each of these problems. You do not

need to solve them.

a

a Chairs cost $Chairs cost $cc at a store. The cost of 6 chairs is $546. at a store. The cost of 6 chairs is $546.

b

b Patrick works forPatrick works for x x hours each day. In a 5-day working week, hours each day. In a 5-day working week,

he works 37

he works 3711

2

2 hours in total. hours in total.

c

c Pens cost $Pens cost $aa each and pencils cost $ each and pencils cost $bb. Twelve pens and three. Twelve pens and three

pencils cost $28 in total.

d

d Amy isAmy is f fyears old. In 10 years’ time her age will be 27.years old. In 10 years’ time her age will be 27.

e

e Andrew’s age isAndrew’s age is j j and Hailey’s age is and Hailey’s age ismm. In 10 years’ time. In 10 years’ time

their combined age will be 80.

their combined age will be 80. Example 3b–d

Example 3b–d

W

W O O R RK K I I N N G G M M A A T T H H _E_E M

(8)

11

11 Find a value ofFind a value of mm that would make this equation true: 10 that would make this equation true: 10 ==_m_m ++_7._7.

12

12 Find two possible values ofFind two possible values of k k that would make this equation true: that would make this equation true: k k ××₍₈₍₈−−_k_k₎₎ ==_12._12.

13

13 If the equationIf the equation x x ++_y_y==_{6 is true, and}_{6 is true, and}_x_x_and_and_y_y_{are both}_{are both}_{whole numbers between 1 and 5,}_{whole numbers between 1 and 5,}_{what values}_{what values}

could they have?

14

14 Equations invEquations involving pronumerals can be olving pronumerals can be split into three groups:split into three groups:

A: Always true, no matter what values are substituted.

N: Never true, no matter what values are substituted.

S: Sometimes true but sometimes false, depending on the values substituted.

Categorise each of these equations as either A, N or S.

a a x x ++₅₅==₁₁₁₁ _b_b ₁₂₁₂−−_x_x==_x_x _c_c _a_a==_a_a _d_d ₅₅++_b_b==_b_b ++₅₅ e e aa==_a_a ++₇₇ _f_f ₅₅++_b_b ==_b_b−−₅₅ _g_g ₀₀××_b_b ==₀₀ _h_h _a_a ××_a_a==₁₀₀₁₀₀ i i 22 x x ++_x_x==₃₃_x_x _j_j ₂₂_x_x++_x_x==₄₄_x_x _k_k ₂₂_x_x++_x_x==₃₃_x_x++₁₁ _l_l _a_a ××_a_a ++₁₀₀₁₀₀==₀₀ W

Enrichment:

Equation permutations

15

15 For each of the following, rearrange the symbols to make a true equation.For each of the following, rearrange the symbols to make a true equation.

a

a 6, 2, 3,6, 2, 3,××_,_,== _b_b _{1, 4, 5,}_{1, 4, 5,}−−_,_,==

c

c 2, 2, 7, 10,2, 2, 7, 10,−−_,_, ÷÷_,_, == _d_d _{2, 4, 5, 10,}_{2, 4, 5, 10,} −−_,_, ÷÷_,_, ==

16

16 aa How many different equatiHow many different equations can be produced ons can be produced using the symbols 2, 3, using the symbols 2, 3, 5,5,++_,_, ==_?_?

b

b How many of these equations are true?How many of these equations are true?

c

c Is it possible to Is it possible to change just one of the change just one of the numbers above and still produce true equations bynumbers above and still produce true equations by

rearranging the symbols?

d

d Is it possible to Is it possible to change just the operation above (i.e.change just the operation above (i.e. ++_{) and still produce true}_{) and still produce true}_equations?_equations?

Many mathematical equations need to be solved in order to build and

launch space stations into orbit.

W

(9)

Solving equations by inspection

Solving an equation is the process of

Solving an equation is the process of ﬁnding the values that pronumerals must take in order to make theﬁnding the values that pronumerals must take in order to make the

equation true. Pronumerals are also

equation true. Pronumerals are also calledcalledunknownsunknowns when solving equations. For simple equations, it when solving equations. For simple equations, it

is possible to ﬁnd

is possible to ﬁnd a solution by trying a a solution by trying a few values for the pronumeral until the equation is true. Thisfew values for the pronumeral until the equation is true. This

method does not guarantee that we have found all the

method does not guarantee that we have found all the solutions (if there is more than one) solutions (if there is more than one) and it will notand it will not

help if there are no

help if there are no solutions, but it can be a solutions, but it can be a useful and quick method for useful and quick method for simple equations.simple equations.

Let’s start:

Finding the missing value

•

• Find the missing values to make theFind the missing values to make the

following equations true.

10 10×× −−₁₇₁₇==₁₃₁₃ 27 27==₁₅₁₅ ++₃₃×× 2 2×× ++₄₄==₁₇₁₇ ++ •

• Can you always ﬁnd a value to put inCan you always ﬁnd a value to put in

the

the place place of of in in any any equation?equation?

9B

■

■ SolvingSolving an equation means ﬁnding the values of any pronumerals to make the equation true. an equation means ﬁnding the values of any pronumerals to make the equation true. These values are called

These values are called solutionssolutions..

■

■ AnAnunknownunknown in an equation is a pronumeral whose value needs to be found in order to make the in an equation is a pronumeral whose value needs to be found in order to make the equation true.

equation true.

■

■ One method of solving equations is One method of solving equations is bybyinspectioninspection (also called (also called trial and errortrial and error or guess andor guess and check), which involves inspecting (or trying) different values and seeing which ones make the

check), which involves inspecting (or trying) different values and seeing which ones make the

equation true. equation true. K K e e y y i i d d e e

a a s s

Example 4

Finding the missing number

For each of these equations, ﬁnd the value of the missing number that would make it true.

a

a ××₇₇==₃₅₃₅ _b_b ₂₀₂₀ −− ==₁₄₁₄

S

a

a 55 55××₇₇==_{35 is a}_{35 is a}_{true equation.}_{true equation.}

b

(10)

Example 5

Solving equations

Solve each of the following equations by inspection.

a a cc ++₁₂₁₂ ==₃₀₃₀ _b_b ₅₅_b_b ==₂₀₂₀ _c_c ₂₂_x_x++₁₃₁₃==₂₁₂₁ _d_d _x_x22 ==₉₉ S SOOLLUUTTIIOONN EEXXPPLLAANNAATTIIOONN a a cc++₁₂₁₂==₃₀₃₀ c c==₁₈₁₈

The unknown variable here is

The unknown variable here is cc..

18

18++₁₂₁₂==_{30 is a}_{30 is a}_{true equation.}_{true equation.}

b

b 55bb ==₂₀₂₀ b b ==₄₄

The unknown variable here is bb..

Recall that 5

Recall that 5bb means 5 means 5 ××_b_b_{, so if}_{, so if}_b_b ==_4,_4,

5 5bb==₅₅××₄₄==_20._20. c c 22 x x ++₁₃₁₃ ==₂₁₂₁ x x ==₄₄

The unknown variable here is x x ..

Trying a few values:

Trying a few values: x

x ==_{10 makes LHS}_{10 makes LHS} ==₂₀₂₀ ++₁₃₁₃==_{33, which is too large.}_{33, which is too large.} x

x ==_{3 makes LHS}_{3 makes LHS}==₆₆++₁₃₁₃==_{19, which is too small.}_{19, which is too small.} x x ==_{4 makes LHS}_{4 makes LHS}==_21._21. d d x x 22 ==₉₉ x x ==−−_3,_3,_x_x==₃₃ x x ==±±₃₃ (

(−−₃₎₃₎22==_{9 is a}_{9 is a}_{true equation}_{true equation}

and

(3)

(3)22==_{9 is also a}_{9 is also a}_{true equation.}_{true equation.}

This equation has two

This equation has two solutions.solutions.

1

1 If the missing number is If the missing number is 5, classify each of the 5, classify each of the following equatifollowing equations as true or ons as true or false.false.

a

a ++₃₃==₈₈ _b_b ₁₀₁₀×× ++₂₂==₄₆₄₆

c

c 1010−− ==₅₅ _d_d ₁₂₁₂==₆₆++ ××₂₂

2

2 For the equationFor the equation ++₇₇==_13:_13:

a

a Find the value of the LHS (left-hand side) ifFind the value of the LHS (left-hand side) if ==_5._5.

b

b Find the value of the LHS ifFind the value of the LHS if ==_10._10.

c

c Find the value of the LHS ifFind the value of the LHS if ==_6._6.

d

d What What value value of of would would make make the the LHS LHS equal equal to to 13?13?

3

3 Find the value of the missing numbers.Find the value of the missing numbers.

a a 44++ ==₇₇ _b_b ₂₂×× ==₁₂₁₂ c c 1313== ++₃₃ _d_d ₁₀₁₀==₆₆++ e e 4242== ××₇₇ _f_f ₁₀₀₁₀₀−− ==₃₀₃₀ g g 1515++₆₆== ++₁₁ _h_h ++₁₁₁₁==₄₉₄₉−− 4

4 Name the unknown pronumeral in each of the following equations?Name the unknown pronumeral in each of the following equations?

a a 44++ ==₁₂₁₂ _b_b ₅₀₅₀−− ==₃₃ Example 4 Example 4

Exercise 9B

W

(11)

5

5 Solve the following equations by inspection.Solve the following equations by inspection.

a a 88××_y_y==₆₄₆₄ _b_b ₆₆÷÷��==₃₃ _c_c ��××₃₃==₁₈₁₈ d d 44−−_d_d==₂₂ _e_e �� ++₂₂==₁₄₁₄ _f_f _a_a−−₂₂==₄₄ g g ss++₇₇==₁₉₁₉ _h_h _x_x÷÷₈₈==₁₁ _i_i ₁₂₁₂==_e_e++₄₄ j j r r ÷÷₁₀₁₀==₁₁ _k_k ₁₃₁₃ ==₅₅++_s_s _l_l ₀₀==₃₃−−_z_z 6

6 Solve the following equations by inspection.Solve the following equations by inspection.

a a 22 p p−−₁₁==₅₅ _b_b ₃₃_p_p++₂₂==₁₄₁₄ _c_c ₄₄_q_q −−₄₄==₈₈ d d 44vv ++₄₄==₂₄₂₄ _e_e ₂₂_b_b−−₁₁==₁₁ _f_f ₅₅_u_u ++₁₁==₂₁₂₁ g g 55gg ++₅₅==₂₀₂₀ _h_h ₄₍₄₍_e_e−−₂₎₂₎==₄₄ _i_i ₄₅₄₅==₅₍₅₍_d_d++₅₎₅₎ j j 33d d −−₅₅==₁₃₁₃ _k_k ₈₈==₃₃_m_m−−₄₄ _l_l ₈₈==₃₃_o_o −−₁₁ 7

7 Solve the following equations by inspection. (All solutions are whole numbers between 1 and 10.)Solve the following equations by inspection. (All solutions are whole numbers between 1 and 10.)

a a 44××₍₍_x_x++₁₎₁₎−−₅₅==₁₁₁₁ _b_b ₇₇++_x_x==₂₂××_x_x _c_c ₍₃₍₃_x_x++₁₎₁₎÷÷₂₂==₈₈ d d 1010−−_x_x==_x_x++₂₂ _e_e ₂₂××₍₍_x_x++₃₎₃₎++₄₄==₁₂₁₂ _f_f ₁₅₁₅−−₂₂_x_x==_x_x g g x x 22==₄₄ _h_h _x_x22==₁₀₀₁₀₀ _i_i ₃₆₃₆==_x_x22 Example 5a,b Example 5a,b Example 5c Example 5c Example 5d Example 5d W W M M A A T T H H _E_E M

L L Y Y U U FF R R PSPS C C 8

8 Find the value of the number in each of these examples.Find the value of the number in each of these examples.

a

a A number is doubled and the result is 22.A number is doubled and the result is 22.

b

b 3 less than a number is 9.3 less than a number is 9.

c

c Half of a number is 8.Half of a number is 8.

d

d 7 more than a number is 40.7 more than a number is 40.

e

e A number is divided by 10, giving a result of 3A number is divided by 10, giving a result of 3

f

f 10 is divided by a number, giving a result of 5.10 is divided by a number, giving a result of 5.

9

9 Justine is paid $10 an hour forJustine is paid $10 an hour for x x hours. During a particular week, she earns $180. hours. During a particular week, she earns $180.

a

a Write an equation involvingWrite an equation involving x x to describe this situation. to describe this situation.

b

b Solve the equation by inspection to ﬁnd the value ofSolve the equation by inspection to ﬁnd the value of x x ..

10

10 Karim’s weight isKarim’s weight is ww kg and his brother is twice as heavy, weighing 70 kg. kg and his brother is twice as heavy, weighing 70 kg.

a

a Write an equation involvingWrite an equation involving ww to describe this situation. to describe this situation.

b

b Solve the equation by inspection to ﬁnd the Solve the equation by inspection to ﬁnd the value ofvalue ofww..

11

11 TTaylah buyaylah buyss x x kg of apples at $4.50 kg of apples at $4.50 per kg. Sheper kg. She

spends a total of $13.50.

a

a Write an equation involvingWrite an equation involving x x to describe this to describe this

situation.

b

b Solve the equation by inspection to ﬁndSolve the equation by inspection to ﬁnd x x ..

W

(12)

12

12 Yanni’s current age isYanni’s current age is y y years old. In 12 years’ time he will be three times as old. years old. In 12 years’ time he will be three times as old.

a

a Write an equation involvingWrite an equation involving y y to describe this situation. to describe this situation.

b

b Solve the equation by inspection to ﬁndSolve the equation by inspection to ﬁnd y y..

W

L L Y Y U U FF R R PSPS C C 13

13 aa Solve the equationSolve the equation x x ++₍₍_x_x++₁₎₁₎==_{19 by inspection.}_{19 by inspection.}

b

b The expressionThe expression x x ++₍₍_x_x++_{1) can be simpliﬁed to 2}_{1) can be simpliﬁed to 2}_x_x++_{1. Use this observation to solve}_{1. Use this observation to solve} x

x ++₍₍_x_x++₁₎₁₎==_{181 by inspection.}_{181 by inspection.}

14

14 There are three consecutive whole numbers that add to 45.There are three consecutive whole numbers that add to 45.

a

a Solve the equationSolve the equation x x ++₍₍_x_x++₁₎₁₎++₍₍_x_x++₂₎₂₎==_{45 by}_{45 by}_{inspection to ﬁnd the three numbers.}_{inspection to ﬁnd the three numbers.}

b

b An equation of the formAn equation of the form x x ++₍₍_x_x++₁₎₁₎++₍₍_x_x++₂₎₂₎==_{? has a whole number solution only if the}_{? has a whole number solution only if the}

right-hand side is a multiple of

right-hand side is a multiple of 3. Explain why this is 3. Explain why this is the case. (Hint: Simplify the LHS.)the case. (Hint: Simplify the LHS.)

W

Enrichment:

Multiple variables

15

15 When multiple variables are involved, inspection can still be used to ﬁnd a solution. For each of theWhen multiple variables are involved, inspection can still be used to ﬁnd a solution. For each of the

following equati

following equations ﬁnd, by ons ﬁnd, by inspection, one pair of values forinspection, one pair of values for x x and and y y that make them true. that make them true.

a

a x x ++_y_y==₈₈ _b_b _x_x−−_y_y==₂₂ _c_c ₃₃==₂₂_x_x++_y_y

d

(13)

Equivalent equations

Sometimes, two equations essentially express the same thing. For example, the equations

Sometimes, two equations essentially express the same thing. For example, the equations x x ++₅₅==_14,_14,

x

x ++₆₆==_{15 and}_{15 and}_x_x++₇₇==_{16 are all made true by the same value of}_{16 are all made true by the same value of}_x_x_{. Each time, we have added one to both}_{. Each time, we have added one to both}

sides of the equation.

We can pretend that true equations are about different objects that have the same weight. For instance,

to say that 3

to say that 3 ++₅₅==_{8 means that a 3 kg}_{8 means that a 3 kg}_{block added to a 5 kg block weighs the same as an 8}_{block added to a 5 kg block weighs the same as an 8}_{kg block.}_{kg block.}

double double both sides both sides subtract 3 from subtract 3 from both sides both sides add 1 to add 1 to both sides both sides x x₊₊66 ₌₌1515 2 2 x x₊₊1010₌₌2828 x x₊₊22 ₌₌1111 initial equation initial equation x x₊₊55 ₌₌1414 1 144 1 144 1 144 1144 11 11 x x 1 1 1 1 1 1 1 1 1 1 x x 1 1 1 1 1 1 1 1 1 1 x x 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x x 1 1 1 1 x x 1 1 1 1 1 1 1 1 1 1

A true equation stays true if we ‘do the same thing to both sides’, such as adding a number or

multiplying by a number. The exception to this rule is that multiplying both sides of any equation by

zero will always make the equation true, and dividing both sides of any equation by zero is not permitted

because nothing can be divided by zero. If we do the same thing to both sides we will have an equivalent

equation.

9C

(14)

Let’s start:

Equations as scales

The scales in the diagram show 2

The scales in the diagram show 2 ++₃₃_x_x==_8._8.

•

• What would the scales look like if two ‘1 kg’ blocks wereWhat would the scales look like if two ‘1 kg’ blocks were

removed from both sides?

•

• What would the scales look like if the two ‘1 kg’ blocks wereWhat would the scales look like if the two ‘1 kg’ blocks were

removed only from the left-hand side? (Try to show whether

they would be level.)

•

• Use scales to illustrate why 4Use scales to illustrate why 4 x x ++₃₃==_{4 and 4}_{4 and 4}_x_x==_{1 are}_{1 are}

equivalent equations. equivalent equations. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x x x x x x ■

■ Two equations areTwo equations are equivalentequivalent if you can get from one if you can get from one to the other by repeatedly:to the other by repeatedly:

–

– Adding a number to both sidesAdding a number to both sides –

– Subtracting a number from both sidesSubtracting a number from both sides –

– Multiplying both sides by a number Multiplying both sides by a number other than zeroother than zero –

– Dividing both sides by a number other Dividing both sides by a number other than zerothan zero –

– Swapping the left-hand side with the right-hand side of Swapping the left-hand side with the right-hand side of the equationthe equation

K K e e y y i i d d e e

a a s s

Example 6

Applying an operation

For each equation, ﬁnd the result of applying the given operation to both sides and then simplify.

a

a 22++_x_x==₅₅ _[add_[add₄₄_to_to_both_both_sides]_sides] _b_b ₇₇_x_x==₁₀₁₀ _[multiply_[multiply_both_both_sides_sides_by_by_2]_2]

c

c 3030==₂₀₂₀_b_b _[divide_[divide_both_both_sides_sides_by_by_10]_10] _d_d ₇₇_q_q−−₄₄==₁₀₁₀ _[add_[add₄₄_to_to_both_both_sides]_sides]

S SOOLLUUTTIIOONN EEXXPPLLAANNAATTIIOONN a a 22++_x_x==₅₅ 2 2++_x_x++₄₄==₅₅++₄₄ x x ++₆₆==₉₉

The equation is written out, and 4 is added to both sides.

Simplify the expressions on each

Simplify the expressions on each side.side.

b b 77 x x ==₁₀₁₀ 7 7 x x ××₂₂==₁₀₁₀××₂₂ 14 14 x x ==₂₀₂₀

The equation is written out, and both

The equation is written out, and both sides are multiplied by 2.sides are multiplied by 2.

c c 3030 ==₂₀₂₀_b_b 30 30 10 10 20 20 10 10 = = b b 3 3==₂₂_b_b

The equation is written out, and both

The equation is written out, and both sides are divided by 10.sides are divided by 10.

d d 77qq−−₄₄==₁₀₁₀ 7 7qq −−₄₄++₄₄==₁₀₁₀ ++₄₄ 7 7qq==₁₄₁₄

The equation is written out, and 4 is added to both sides.