australian curriculum mathematics year 7mathematics for the following year 7 content descriptions....

AUSTRALIAN CURRICULUM

MATHEMATICS YEAR 7

Solving equations

MATHEMATICS YEAR 7

Solving equations

Student’s name: ________________________________

Teacher’s name: ________________________________

First published 2012

ISBN 9780730744412

SCIS 1564084

© Department of Education WA 2012 (Revised 2020)

Requests and enquiries concerning copyright should be addressed to:

Manager Intellectual Property and Copyright Department of Education 151 Royal Street EAST PERTH WA 6004

Email: [email protected]

This resource contains extracts from The Australian Curriculum Version 3.0 © Australian Curriculum, Assessment and Reporting Authority 2012. ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that:

the content descriptions are solely for a particular year and subject

all the content descriptions for that year and subject have been used

the author’s material aligns with the Australian Curriculum content descriptions for the relevant year andsubject.

You can find the unaltered and most up to date version of this material at www.australiancurriculum.edu.au. This material is reproduced with the permission of ACARA.

creativecommons.org/licenses/by-nc-sa/3.0/au/

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Year 7 Mathematics Solving equations

© Department of Education WA 2012 – MATHSAC021 Page 1

Contents

Signposts ....................................................................................................................................2

Introduction...............................................................................................................................3

Curriculum details....................................................................................................................4

1. Integer operations ...............................................................................................................7

2. Algebraic expressions .......................................................................................................11

3. Equations ...........................................................................................................................15

4. Solving equations 1 ...........................................................................................................21



7. Using equations .................................................................................................................37

8. Summary...........................................................................................................................43

9. Review tasks ......................................................................................................................45

Solutions...................................................................................................................................51

Solving equations Year 7 Mathematics

Page 2 © Department of Education WA 2012 – MATHSAC021

Signposts

Each symbol is a sign to help you.

Here is what each one means:

The recommended time you should take to complete this section.

An explanation of key terms, concepts or processes.

A written response. Write your answer or response in your journal.

Correct this task using the answers at the end of the resource.

Calculators may not be used here.

Make notes describing how you attempted to solve the problem. Keep these notes to refer to when completing the Self-evaluation task. Your teacher may wish you to forward these notes.



Introduction

This resource should take you approximately two weeks to complete. It comprises seven learning sections, a summary section and a review task section.

The learning sections have the following headings:

Key wordsThese are the main words that you need to understand and use fluently to explain yourthinking.

Warm-upWarm-up tasks should take you no longer than 10 minutes to complete. These are skillsfrom previous work you are expected to recall from memory, or mental calculations thatyou are expected to perform quickly and accurately. If you have any difficulties inanswering these questions, please discuss them with your teacher.

ReviewSome sections have reviews immediately after the warm-up. The skills in these reviewsare from previous work and are essential for that section. You will use these to developnew skills in mathematics. Please speak to your teacher immediately if you are havingany trouble in completing these activities.

Focus problemFocus problems are designed to introduce new concepts. They provide examples of thetypes of problems you will be able to solve by learning the new concepts in this resource.Do not spend too long on these but do check and read the solutions thoroughly.

Skills developmentThese help you consolidate new work and concepts. Most sections include skillsdevelopment activities which provide opportunities for you to become skilled at usingnew procedures, apply your learning to solve problems and justify your ideas. Pleasemark your work after completing each part.

Correcting your work

Please mark and correct your work as you go. Worked solutions are provided to show how you should set out your work. If you are having any difficulty in understanding them, or are getting the majority of the questions wrong, please speak to your teacher immediately.

Journal

Please keep an exercise book to record your notes and to summarise your learning. At the end of each section, write definitions for the key words that were introduced for that section.



Curriculum details Content Descriptions This resource provides learning and teaching to deliver the Australian Curriculum: Mathematics for the following Year 7 Content Descriptions.

Compare, order, add and subtract integers (ACMNA280)

Extend and apply the laws and properties of arithmetic to algebraic terms and expressions (ACMNA177)

Solve simple linear equations (ACMNA179)

Content Descriptions 1 2 3 4 5 6 7 R

ACMNA 280

ACMNA 177

ACMNA179

Indicates the content description is explicitly covered in that section of the resource.

Previous relevant Content Descriptions

The following Content Descriptions should be considered as prior learning for students using this resource.

At Year 5 level

Use equivalent number sentences involving multiplication and division to find unknown quantities (ACMNA121)

Proficiency strand statements at Year 7 level At this year level:

Understanding includes describing patterns in uses of indices with whole numbers, recognising equivalences between fractions, decimals, percentages and ratios, plotting points on the Cartesian plane, identifying angles formed by a transversal crossing a pair of lines, and connecting the laws and properties of numbers to algebraic terms and expressions

Fluency includes calculating accurately with integers, representing fractions and decimals in various ways, investigating best buys, finding measures of central tendency and calculating areas of shapes and volumes of prisms

Problem Solving includes formulating and solving authentic problems using numbers and measurements,working with transformations and identifying symmetry, calculating angles and interpreting sets of data collected through chance experiments



Reasoning includes applying the number laws to calculations, applying known geometric facts to draw conclusions about shapes, applying an understanding of ratio and interpreting data displays

General capabilities General capabilities 1 2 3 4 5 6 7 R

Literacy

Numeracy

Information and communication technology (ICT) capability

Critical and creative thinking

Personal and social capability

Ethical behaviour

Intercultural understanding

Indicates general capabilities are explicitly covered in that section of the resource.

Cross-curriculum priorities Cross-curriculum priorities 1 2 3 4 5 6 7 R

Aboriginal and Torres Strait Islander histories and cultures

Asia and Australia’s engagement with Asia

Sustainability

Indicates cross-curriculum priorities are explicitly covered in that section of the resource.

This resource contains extracts from The Australian Curriculum Version 3.0 © Australian Curriculum, Assessment and Reporting Authority 2012. ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that:

the content descriptions are solely for a particular year and subject

all the content descriptions for that year and subject have been used

the author’s material aligns with the Australian Curriculum content descriptions for the relevant year and subject.

You can find the unaltered and most up to date version of this material at www.australiancurriculum.edu.au. This material is reproduced with the permission of ACARA.

creativecommons.org/licenses/by-nc-sa/3.0/au/



1. Integer operations

When you complete this section you should be able to:

add and subtract integers.

Key words

integer opposite

Warm-up 1

1. Circle the factors of 9. 2, 6, 7, 9, 18

2. 5 + 6 = _________

3. What is the missing number?

a = _________

4. Circle the greater fraction. 2

4 or

1

3

5. What is a half 18? _________

6. 7.1 + 8.5 = _________

7. 5.7 3 = _________

8. Write 0.3 as a fraction. _________

9. Complete by writing in the next number: 75, 83, 91, _________

10. At what point is the truck?

_________

-6 -4 -2 0a

x1 2 3 4 5

y

12345



Review 1.1

Example Complete the following additions of integers.

1. 7 + (-4)

2. (-3) + (-8)

Solution 1. 7 + (-4) = 3

This can be represented as the following.

7: + + + + + + +

(-4): − − − −

It can be seen from this diagram that the four negatives will cancel with four positives, resulting in three positives being left. Hence the result is 3.

2. (-3) + (-8) = (-11)

This can be represented as the following.

(-3): − − −

(-8): − − − − − − − −

It can be seen from this diagram that nothing will cancel out. Combining the two numbers will result in there being 11 negatives. Hence the result is (-11).


1. Evaluate these using whatever working is necessary.

(a) (-5) + 2 ______________ (b) (-4) + (-4) ______________

(c) 9 + (-6) ______________ (d) (-3) + 8 ______________

(e) (-8) + 6 ______________ (f) (-10) + (-6) ______________

(g) 11 + (-4) ______________ (h) (-12) + 12 ______________

(i) (-15) + 8 ______________ (j) (-7) + (-8) ______________

(k) 15 + (-7) ______________ (l) (-13) + 6 ______________

(m) (-13) + 9 ______________ (n) (-7) + (-7) ______________

(o) 16 + (-10) ______________ (p) (-14) + 20 ______________



Review 1.2

Example Rewrite these subtractions so that the opposite is being added, then complete the addition.

1. 8 − (-4)

2. (-3) − (-6)

3. (-8) − 5

Solution 1. 8 − (-4) = 8 + 4 = 12

2. (-3) − (-6) = (-3) + 6 = 3

3. (-8) − 5 = (-8) + (-5) = (-13)


1. Evaluate these integer subtractions using whatever working is necessary.

(a) (-5) − 2 ______________ (b) (-6) − (-6) ______________

(c) 9 − (-4) ______________ (d) (-6) − 8 ______________

(e) (-8) − 5 ______________ (f) (-8) − (-5) ______________

(g) 11 − (-5) ______________ (h) (-12) − 12 ______________

(i) (-19) − 9 ______________ (j) (-7) − (-8) ______________

2. Evaluate these using whatever working is necessary.

(a) 18 − (-7) ______________ (b) 13 + (-6) ______________

(c) 14 + (-21) ______________ (d) (-12) − 6 ______________

(e) (-7) − 11 ______________ (f) (-8) − 6 ______________

(g) (-7) + (-7) ______________ (h) (-8) + 16 ______________

(i) (-7) − (-7) ______________ (j) (-12) + 11 ______________

(k) 21 − (-5) ______________ (l) (-13) − 13 ______________

Check your work before continuing.



2. Algebraic expressions


manipulate algebraic terms and expressions.

Key words

expression simplify expand factorise like

Warm-up 2

1. Circle the common factor of 2 and 5. 1, 2, 3, 4, 5

2. 12 – 4 = _________

3. The temperature was 3 degrees but it dropped 6 degrees.

What is the new temperature? _________

4. Insert <, > or = to make the following sentence true. 2 3

3 4

5. 2

218 = _________

6. Round 4.2 to a whole number. _________

7. 3 30.96

8. Write 331

3% as a decimal. _________

9. Complete by writing in the next number: 44.2, 44.4, 44.6, _________

10. Determine the probability the spinner herewill land on a number greater than 1.Express your answer as a fraction.

_________

1

23

4

0

5

10

20

°C



Review 2.1

Simplify means to write an expression in an equivalent but simpler form. It may have fewer terms after you have simplified it. eg Simplifying 3 8x x x to 11x

Example Use the distributive property and commutative property to simplify this expression:

5 3 4 9x y x y

Solution 5 3 4 9 Note that means 1

5 4 3 9 Commutative property

(5 4) (3 1) 9 Distributive property

9 2 9

x y x y y y

x x y y

x y

x y

This is often called collecting like terms. Like terms are ones with equivalent variable parts

such as 5x and 4x, 3y and y, 3ab and 7ab, 22x and 26 .x If terms are not like it would not be possible to use the distributive property to combine them.

1. Simplify these expressions by collecting like terms.

(a) 4 8x x ________________________

(b) 6 3 3y y y x ________________________

(c) 2 8 3ab ab bc ________________________

(d) 5 4 4 5x y x y ________________________

(e) 23 5 5 7x y x x y ________________________



Review 2.2

Expand means to write without brackets. eg Expanding 4( 2)x to 4 8x

Example Use the distributive property to expand this expression.

3( 5)x

Solution 3( 5)

3 3 5 Distributive property

3 15

x

x

x

1. Expand these expressions.

(a) 4( 5)x ________________________

(b) 6( 2)y ________________________

(c) 5(8 )z ________________________

(d) 3( 5)x y ________________________

(e) 3( 3)ab ________________________

Review 2.3

Factorising means writing an expression as a product of factors. eg Factorising 4 8x to 4( 2)x Note that written as 4( 2)x it has been written as a product of the factors 4 and 2.x

Example Use the distributive property to factorise this expression.

5 25x

Solution 5 25

5 5 5

5( 5)

x

x

x



1. Factorise these expressions.

(a) 6 9a ________________________

(b) 4 8b ________________________

(c) 20 5c ________________________

(d) 18 18d ________________________

(e) 3 6xy ________________________




3. Equations


write linear equations recognise equivalent equations.

Key words

variable equation substitute substitution equivalent equivalent equation number sentence

Warm-up 3

1. Circle the prime number. 20, 21, 34, 37, 45

2. 8 7 = _________


a = _________

4. Locate 3

5 on the number line.

5. Find a quarter of 16. _________

6. Estimate the sum by first rounding to whole numbers. 8.9 + 9.7 _________

7. 3 . 4

5_________

8. Write 1

5 as a percentage. _________

9. Complete by writing in the next number:1 5 9

7 7 7, , , _________

10. Determine the size of the missing angle.

_________

-6-8 -2 0a

0 1

? 127°



Review 3

Example If a = 3, b = 5 and c = 8, which of these equations are true?

1. a b c

2. c a b

3. a c b

Solution Substitution gives the following.

1. 3 + 5 = 8 True

2. 8 − 3 = 5 True

3. 3 − 8 = 5 False

1. If x = 5, y = 9 and z = 4, which of these are true?

(a) x y z ________ (b) x z y ________

(c) y z x ________ (d) x y z ________

(e) x z y ________ (f) y x z ________

(g) y z x ________ (h) z x y ________

(i) z y x ________

2. Write down three pairs of numbers that add to each of these values.

(a) 10 ________ ________ ________

(b) 5 ________ ________ ________

(c) 20 ________ ________ ________

(d) 0 ________ ________ ________

(e) (-4) ________ ________ ________




Focus problem 3

Sum Shapes 1. In the diagram here find numbers to place in the circles so that

the number in each square is the sum of the numbers in the twocircles attached to it.

2. Now find another set of numbers to place in the circles.

_______________________________________________________________________

_______________________________________________________________________

3. The diagram here has used the variables a to h to represent thenumbers in the shapes.Complete this table to show all of the possible (non-negative)solutions for the puzzle. If your solutions for questions 1 and 2were non-negative, you should find them in the table.

a c f h

0 1

1

2 3 3

3

4 1 4 5

5

4. Draw up a table that shows all of the possible non-negative solutions for when the SumShape has b = 3, d = 4, e = 9 and g = 10.



For the left hand side of the Sum Shape here, the following number sentences can be written.

(i) 5 + 3 = 8

(ii) 3 + 5 = 8

(iii) 8 − 3 = 5

(iv) 8 − 5 = 3

These number sentences are said to be equivalent.

5. Write four equivalent number sentences that can be written about the top side of the SumShape.

_______________________________________________________________________

_______________________________________________________________________

For the left hand side of the Sum Shape above, the following equations can be written. (i) a f d

(ii) f a d

(iii) d f a

(iv) d a f

These equations are said to be equivalent.

6. The equation f d a is not a true equation for the Sum Shape. Show it is false bysubstituting the numbers from the Sum Shape into the equation.

_______________________________________________________________________

_______________________________________________________________________

7. Write four equivalent equations that can be written about the top side of the Sum Shape.

_______________________________________________________________________

_______________________________________________________________________

The equation 3a f defines one of the rules for this particular Sum

Shape. That is that the number in each square is the sum of the numbers in the two circles attached to it.



8. Write at least twelve other equations that must be true for this Sum Shape. (Rememberto substitute some correct values if you need to check whether the equations are true.)

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

9. Divide your list of twelve equations into sets of equivalent equations. These contain thesame numbers, arranged into a different order or form.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

10. Rewrite the equation 10x y into as many equivalent forms as you can.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________




Skills development 3

Example For the values x = 3, y = 8 and z = 5 write three equations using x, y and z which are true.

Solution , ,x z y y z x y x z

1. (a) Find values for a, b and c that make this equation true: .a b c

__________________________

(b) Write three other equations equivalent to a b c that are true for your values ofa, b and c.

____________________________________________________

2. (a) Find values for p, q and r that make this equation true: .p r q

__________________________

(b) Write three other equations equivalent to p r q that are true for your values of p,q and r.

____________________________________________________

3. (a) Find values for x, y and z that make this equation true: .z x y

__________________________

(b) Write three other equations equivalent to z x y that are true for your values of x,y and z.

____________________________________________________




4. Solving equations 1


solve equations using a guess and check method.

Key words

equation solve solution substitute integer

Warm-up 4

1. Complete the pattern by writing in the next number: 21, 15, 10, 6, _________

2. 56 7 = _________

3. The temperature was minus 7 degrees but it went up 3 degrees.

What is the new temperature? _________

4. Express the value of w as a fraction. _________

5. 2

315 = _________

6. 0.032 100 = _________

7. 95.49 9 = _________

8. Write 5% as fraction. _________

9. Complete by writing in the next number: 152, 148, 144, ________

10. The truck is at (0, 3).If the truck moves 2 units down,where will it then be?

_________

x1 2 3 4 5

y

12345

10

w



Review 4

Example Substitute x = 3 and y = 6 into the expression 4 2 9x y to find its value.

Solution 4 × 3 − 2 × 6 + 9 = 12 − 12 + 9 = 9

1. Substitute a = 6 into the following expressions to find their value.

(a) a + 5 ________________________

(b) 4a ________________________

(c) 2a ________________________

(d) a − 8 ________________________

(e) 3a + 5 ________________________

2. Substitute b = 4 and c = 5 into the following expressions to find their value.

(a) b + c ________________________

(b) bc ________________________

(c) 4b + c ________________________

(d) c

b________________________

(e) 5 − b + 3c ________________________

3. Substitute d = 5 and e = 9 into the following expressions to find their value

(a) 6e

d

________________________

(b) ( 4)d e ________________________

(c) 2 2 8e d ________________________

(d) 23d ________________________

(e) 2e d ________________________




Focus problem 4

The square of a number is twelve more than four times the number. Find the number.

One way of solving the puzzle is to write an equation for it, and then substitute various values until the equation is solved.

The equation for this puzzle is: 2 12 4x x

When the values of the solutions are substituted in for x, the equation will be true. There are two of these values (solutions) for this equation. Hence, there are two numbers that solve the puzzle.

Here is an example with (-4) substituted into the equation.

2

2

12 4

(-4) 12 4 (-4)

16 12 (-16)

4 (-16) is false.

x x

As substituting (-4) makes the equation false, (-4) is not a solution.

Substitute each of the integers: -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 into the equation to find the two solutions. It may help to use a calculator, especially with the negative integers.


Solutions to equations

The number of solutions to this type of equation is the highest power of the variable.

As the equation 2 12 4x x has 2 as the highest power, there will be two solutions.

The equation 5 9 24x has a highest power of one, so there will only be one solution.




Example Solve the equation 3 5 14x using the ‘guess and check’ method.

Solution Try x = 1: 3 1 5 14 False so 1 Need a larger value of x x

Try x = 2: 3 2 5 14 False so 2 Need a larger value of x x

Try x = 5: 3 5 5 14 False so 5 Need a smaller value of x x

Try x = 3: 3 3 5 14 True so 3x

1. Solve the following equations using the guess and check method. With the easierquestions, you may be able to work out the solution straight away.

(a) 5 13x ______________

(b) 7 4x ______________

(c) 15 5x ______________

(d) 4 20x ______________

(e) 63

x ______________

(f) 17 8 x ______________

(g) 6

2x ______________

(h) 2 3 11x ______________

(i) 3 5 7x ______________

(j) 8 2 ( 2)x ______________

(k) 15 3 12x ______________

(l) 6 13 1x ______________






solve equations using backtracking.

Key words

equation solve solution substitution

Warm-up 5

1. Circle the square number. 2, 4, 6, 8

2. 217 + 28 = __________


c = __________

4.3 1

5 5 __________

5. What is a tenth of 80? __________

6. 86 cm = __________ m

7. 9 + 7 2 = __________

8. Write 2

3as decimal. __________

9. Complete by writing in the next number: 0.09, 0.07, 0.05, __________

10. A six-sided die is rolled.Express, as a percentage, the probabilitythat it lands on a number less than 7.

__________

-0-9 -3 3c



Review 5

Example Write the inverse operation for these.

1. + 7

2. − 5

3. × 4

4. ÷ 6

Solution 1. − 7

2. + 5

3. ÷ 4

4. × 6

1. Write the inverse operations for these.

(a) + 9 ______

(b) − 3 ______

(c) × 5 ______

(d) ÷ 8 ______

(e) − 10 ______

(f) + 12 ______

(g) ÷ 7 ______

(h) × 6 ______




Focus problem 5

Think of a number.

Double that number.

Add five to the result.

Multiply the result by two.

Subtract 10 from the result.

Divide the result by four.

1. What do you notice?

________________________________________________________

2. Try the process again, this time starting with a different number. What do you notice?

________________________________________________________

3. Try to explain why this would happen, no matter what number you started with.

________________________________________________________

________________________________________________________

________________________________________________________

________________________________________________________


Number puzzles

Number puzzles like the one in Focus problem 5 are usually created using algebra. In this way the writer can always be sure the result is the same as the starting number, no matter what starting number is chosen.




Example The equation 2 5 13x represents starting with a number, x, multiplying it by 2 and then adding 5 to give a result of 13. Show how using the inverse operations, applied in the reverse of the original order, can solve the equation.

Solution Start at the result of 13.

Then reverse the operation of adding 5 by using the inverse operation, subtracting 5, ie 13 − 5 = 8.

Then reverse the operation of multiplying by 2 by using the inverse operation, dividing by 2, ie 8 ÷ 2 = 4.

This means the starting number, x, was 4.

Checking: 2 × 4 + 5 = 8 + 5 = 13 so x = 4 is the correct solution.

Here is a diagram that summarises the process used in the example.

As well as using the correct inverse operations, the order must be reversed. If we divided by 2 then subtracted 5 it would not work, as shown here.

13 ÷ 2 = 6.5

6.5 − 5 = 1.5

Checking: 2 × 1.5 + 5 = 3 + 5 = 8 not 13, so 1.5 is not the correct solution.

Reversing the process is usually referred to as backtracking.



1. Solve these equations using backtracking, then check your solutions by substitution.

(a) 3 4 11x ________________________________________________

________________________________________________

________________________________________________

(b) 5 2 17x ________________________________________________

________________________________________________

________________________________________________

(c) 4 8 0x ________________________________________________

________________________________________________

________________________________________________

(d) 12 17x ________________________________________________

________________________________________________

________________________________________________

(e) 3 2 9x ________________________________________________

________________________________________________

________________________________________________






solve equations by finding simpler equivalent equations.

Key words

equation equivalent equation solve solution substitution

Warm-up 6

1. 5.6 10 = _________

2. 357 – 49 = _________

3. The temperature is minus 3 degrees.

How much will it need to increase to get to 4 degrees? _________

4.5 2

5 5 _________

5. 1

440 = _________

6. 550 g = _________ kg

7. 9 – 4 + 1 = _________

8. Write 0.42 as a percentage. __________

9. Complete by writing in the next number: 6 4 2

8 8 8, , , ________

10. Which shape is at (-1,3)?

__________________

x-5 -4 -3 -2 -1 1 2 3 4 5

y

1

2

3

4

5



Review 6

Example Solve the equation 6 8 16x by guess and check and by backtracking.

Solution Guess and check.

Try x = 2: 6 × 2 − 8 = 12 – 8 = 4 → Too small (aiming for 16) so increase x.

Try x = 4: 6 × 4 − 8 = 24 – 8 = 16 → So x = 4 is the correct solution.

Backtracking.

6 8 16x

The x is multiplied by 6 then 8 is subtracted.

To backtrack we need to add 8 then divide by 6. 16 add 8 is 24.

24 divided by 6 is 4, ie x = 4 is the solution.

1. Solve these equations by backtracking.

(a) 2 5 11x ________________________________________________

________________________________________________

________________________________________________

(b) 4 3 13x ________________________________________________

________________________________________________

________________________________________________

2. Solve these equations by guess and check.

(a) 3 4 1x ________________________________________________

________________________________________________

________________________________________________

(b) 3 4 (-5)x ________________________________________________

________________________________________________

________________________________________________




Focus problem 6

Equivalent equations always have the same solutions.

The equations 5 8x and 6 9x are equivalent as they both have the solution x = 3.

1. Use substitution to show that all of these equations are equivalent.

(a) 6 10x ____________________

(b) 10 14x ____________________

(c) 2 2x ____________________

(d) 2 6 14x ____________________

(e) 2 6 10x x ____________________

2. Find three pairs of equivalent equations amongst these equations. One method is to findsolutions to the equations.

(a) 3 6 18x (b) 5 3x

____________________ ____________________

(c) 10 7x (d) 2 1 9x

____________________ ____________________

(e) 6 2x (f) 4 8x

____________________ ____________________

(g) 10 12x (h) 5 2 7x

____________________ ____________________

(i) 5 12 7x (j) 4 9x

____________________ ____________________

____________________ ____________________ ____________________



Equivalent equations can be developed by adjusting a given equation in a balanced way. This means changing both sides of the equation in the same way. When the new equation is written, both sides are still of the same value. They can be said to be balanced.

For example, given that x = 3 the equation 4 2 14x is true, and both sides have the same value of 14.

If 5 was added to each side of this equation it would still be balanced for x = 3 and you would get an equivalent equation.

3. After adding 5 to both sides the new equation is 4 7 19x . Check by substitution thatthis equation is equivalent to the original one, that is, its solution is also x = 3.

________________________________________________

________________________________________________

4. What equivalent equation would you get if you subtracted 2 from both sides of theequation 4 2 14x ?

________________________________________________

________________________________________________

5. What equivalent equation would you get if you divided both sides of the equation4 12x by 4?

________________________________________________

________________________________________________

Notice that in questions 4 and 5 above you have solved the equation 4 2 14x by altering both sides of the equation equally, in two steps. As with backtracking, the operation applied to each side of the equation depends on what operation we are trying to undo, and the order we undo the equation is the reverse of the rule of order.

Here is the equation solved using equivalent equations.

4 2 14

4 2 2 14 2 2 is subtracted from each side

4 12

4 12both sides are divided by 4

4 41 3

ie 3

x

x

x

x

x

x

This shows the full setting out which can be simplified by leaving out obvious steps. Note in solving equations this way, the equal signs are aligned.



6. Solve these two equations setting out your work as shown above.

(a) 3 1 16x (b) 5 6 19x

________________ ________________

________________ ________________

________________ ________________

________________ ________________

________________ ________________

________________ ________________


Optimisation

One of the most significant uses of mathematics is in the process of optimisation. This means finding the solution to an equation that maximises or minimises some quantity. The problem might be maximising profits or minimising costs in business, minimising the air resistance of a new car design or maximising the power from an engine.

All of these optimisation problems involve solving equations.




Example Solve the equation 5 3 (-13)x using equivalent equations.

Solution 5 3 (-13)

5 3 3 (-13) 3 3 is added to each side

5 (-10)

5 (-10)both sides are divided by 5

5 51 (-2)

ie (-2)

x

x

x

x

x

x

1. Solve these equations using equivalent equations. Check your solutions by substitution.

(a) 7 1 15x (b) 4 3 13x

________________ ________________

________________ ________________

________________ ________________

________________ ________________

________________ ________________

(c) 3 9 0x (d) 37 16 90x

________________ ________________

________________ ________________

________________ ________________

________________ ________________

________________ ________________




7. Using equations


write and solve an equation.

Key words

equation equivalent equivalent equation solve solution

Warm-up 7

1. 10.5 10 = _________

2. 54 3 = _________

3. The temperature is 5 degrees.

How much will it need to decrease to get to minus 1 degree? _________

4.1 1

3 6 _________

5. 1

525 = _________

6. 8700 mL = ________ L

7. 10 (5 + 5) = _________

8. Find 10% of $120. ____________

9. Describe the rule for the following pattern. 15, 30, 45, 60, 75, …

______________________________________________________

10. Determine the size of the missing angle.

_________ 138°?



Review 7

Example Write an algebraic expression to represent the following.

The sum of eight and three times a number is doubled.

Solution All of the following expressions are equivalent and correct.

2(8 3 )x

(8 3 ) 2x

2 (3 8)x

2.(3 8)x

1. Write expressions to represent these statements.

(a) Eight more than a number ___________

(b) Five less than a number ___________

(c) I had x songs on my mp3 then I added 18 more. ___________

(d) The result of multiplying 7 by y ___________

(e) Eleven is added to a number before the result is doubled. ___________

(f) The sum of the square of a number and the number itself ___________

(g) A number is divided by 5 ___________

(h) A number is multiplied by four before 6 is added to the result. ___________

2. Write in words what these expressions represent.

(a) a + 8 ________________________________________________

(b) b − 6 ________________________________________________

(c) 9c ________________________________________________

(d) 3( 5)d ________________________________________________




Focus problem 7

The picture on the right shows a set of scales which are in balance. This means that the sum of the values on both sides of the scales is equal.

It is obvious that there is a 2 on the left and a 5 on the right hand side. But the value of the squares, x, is not known.

1. Using a guess and check method, find the value of the x that will balance the scales.

___________

2. Write an expression representing each side of the scale. Use these expressions to write anequation representing the scale is balanced.

____________________________________________

____________________________________________

3. If your equation in question 2 is correct, the following two steps should give you simplerequivalent equations and the solution.

Step 1: Subtract 2 from each side of your equation. ______________________

Step 2: Subtract 1x from each side of your equation. ______________________

4. What is the value of x? Does this agree with your answer from question 1? ___________


Scales of justice

Scales like these have been used historically to represent justice. They represent balancing the ‘for’ and ‘against’ of a case. Many sculptures depict these scales, which are one of three symbols depicted with Lady Justice.




Example Handing over a $20 note to pay for four ice creams results in getting $8 change. Write an equation for this and solve the equation to find the price of an ice cream.

Solution If x is the price of an ice cream:

4 8 20

4 8 8 20 8

4 12

4 12

4 43

x

x

x

x

x

So the price of each ice cream is $3.

1. Write equations for and solve these problems.

(a) Paying for three avocados with a $10 note results in $4 change. How much waseach avocado?

________________________________________________

________________________________________________

________________________________________________

________________________________________________

________________________________________________

________________________________________________

(b) Five identical banknotes and $3 in coins total $28. What is the denomination of thebank notes?

________________________________________________

________________________________________________

________________________________________________

________________________________________________

________________________________________________

________________________________________________



(c) Six cinema tickets cost $78. What is the cost of each ticket?

________________________________________________

________________________________________________

________________________________________________

________________________________________________

________________________________________________

________________________________________________

(d) Cutting 12 cm from a piece of string leaves a length of 20 cm. How long was the stringoriginally?________________________________________________

________________________________________________

________________________________________________

________________________________________________

________________________________________________

________________________________________________




8. Summary

Solving equations means finding the value of the variable that makes the equationtrue.

Three methods of solving equations are (i) guess and check, (ii) backtracking and (iii)equivalent equations.

Guess and check involves guessing a value for the variable, testing it by substitutionand then repeating the process if necessary.

Backtracking involves reversing the order of operations in the equation to return to aunit variable eg 1x.

Equivalent equations involve repeatedly adjusting the equation in the same way onboth sides to arrive at an equivalent equation, until the unit variable is found.

Problems can be solved by first writing an equation to represent the problem thensolving that equation.



9. Review tasksThe following tasks will assist you to consolidate your learning and understanding of the concepts introduced in this resource, and assist you to prepare for assessments.

Task A

Name: _____________________________ Suggested time: 40 minutes

Actual time taken: __________

Instructions

Complete this work on your own.

You may use a calculator, but show how you got your answer.

Attempt every question. Take as long as you need and record the time in the space provided above after you have finished.

1. (a) Find values for x, y and z that make this equation true: .x z y

________________________________________________________

(b) Write three other equations equivalent to x z y that are true for your values of x,

y and z.

________________________________________________________

2. Solve these equations using guess and check method.

(a) 4 5 7x (b) 10 2 (-4)y

________________ ________________

________________ ________________

________________ ________________

________________ ________________

________________ ________________



3. Solve these equations using backtracking.

(a) 5 3 13x (b) 8 3 13x

________________ ________________

________________ ________________

________________ ________________

________________ ________________

________________ ________________

4. Solve these equations using equivalent equations.

(a) 2 3 (-1)x (b) 5 8 23x

________________ ________________

________________ ________________

________________ ________________

________________ ________________

________________ ________________

5. Write equations for and then solve these problems.

(a) Eight identical gold coins weigh 226.8 grams. How much does each coin weigh?

________________________________

________________________________

________________________________

________________________________

________________________________

(b) After buying four blocks of chocolate with a $10 note Jane is given $3 change. Howmuch is each block of chocolate?

________________________________

________________________________

________________________________

________________________________

________________________________



Task B

Name: _____________________________ Suggested time: 40 minutes

Actual time taken: __________

Instructions

Complete this work on your own.

You may use a calculator, but show how you got your answer.

Attempt every question. Take as long as you need and record the time in the space provided above after you have finished.

The image above shows the first three diagrams in a growing sequence of shapes made from squares.

1. Draw diagrams 4 and 5 below.



2. Complete this table of values for the sequence of crosses made from squares.

Diagram x 1 2 3 4 5 6 7

Squares y 5 21

3. By substituting values from your table, select which of these equations correctlyrepresents the values.

3 4y x 4y x y x 5 4y x

3 3y x 4 4y x 4 3y x 2 1y x

________________________________________________________________

4. Use the equation 4 3y x to find out how many squares would be in the diagram

numbered 30.

________________________________________________________________

5. Solve the equation 4 3 49.x

________________________________

________________________________

________________________________

________________________________

________________________________

6. What does solving the equation in question 5 tell you about the crosses from squaressequence?

________________________________________________________________

________________________________________________________________



Self-evaluation task Please complete the following.

How well did you manage your own learning using this resource?

Always Usually Rarely Not sure

Each section took approximately 45 minutes to complete.

I needed extra help.

I marked and corrected my work at the end of each section.

I made the journal entries and summaries when asked.

I have kept to my work schedule.

How much mathematics have you learnt using this resource?

Always Usually Rarely Not sure

Understanding I understand the use of variables in equations and I can represent situations algebraically.

Fluency

I can solve equations.

Problem Solving I solved problems by writing and solving equations by applying various techniques.

Reasoning I can explain the equivalence of two equations algebraically.



Write a list of topics for which you need additional assistance.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________



Solutions

1. Integer operations

Solutions to Warm-up 1

1. 9 is the only factor of 9 in the list.2. 113. (-8)

4.2

45. 96. 15.67. 17.1

8.3

109. 9910. (4, 4)

Solutions to Review 1.1

1. (a) (-3) (b) (-8)(c) 3 (d) 5(e) (-2) (f) (-16)(g) 7 (h) 0(i) (-7) (j) (-15)(k) 8 (l) (-7)(m) (-4) (n) (-14)(o) 6 (p) 6


1. (a) (-7) (b) 0(c) 13 (d) (-14)(e) (-13) (f) (-3)(g) 16 (h) (-24)(i) (-28) (j) 1

2. (a) 25 (b) 7(c) (-7) (d) (-18)(e) (-18) (f) (-14)(g) (-14) (h) 8(i) 0 (j) (-1)(k) 26 (l) (-26)



2. Algebraic expressions


1. 1 should be circled2. 83. (-3°)

4.2 3

3 4

5. 186. 47. 10.328. 0.33…9. 44.8

10.3

4


1. (a) 12x(b) 4 3y x(c) 10 3ab bc(d) 4 9 4x y

(e) 24 7x x


1. (a) 4 20x (b) 6 12y (c) 40 5z(d) 3 3 15x y (e) 3 9ab


1. (a) 3(2 3)a (b) 4( 2)b (c) 5(4 )c(d) 18( 1)d (e) 3( 2)xy



3. Equations


1. 372. 563. (-4)4.

5. 46. 197. 17.08. 20%

9.13

710. 53°

Solutions to Review 3

1. (b), (f), (g)2. Many answers are possible, for example, 5 + 5 = 10, 6 + 4 = 10, 8 + 2 = 10.

Checking with a calculator is worthwhile.

Solution to Focus problem 3

Sum Shapes

1. Here is one possible solution.Others are in the table below for question 3.

2. All possible non-negative solutions are in the table for question 3. Your answer shouldbe one of them.

3.

a c f h

0 5 8 1

1 4 7 2

2 3 6 3

3 2 5 4

4 1 4 5

5 0 3 6



4.

a c f h

0 3 4 6

1 2 3 7

2 1 2 8

3 0 1 9

5. 5 + 7 = 12, 7 + 5 = 12, 12 − 7 = 5, 12 − 5 = 76. 3 − 8 = 5 is false so f d a is not a true equation for the Sum Shape.7. ,a c b ,c a b ,b c a b a c 8. There are 32 possible equations (including the one given, 3a f ). You need to have

written any twelve.Top side Left side Right side Bottom side

9a c 3a f 7h c 1h f

9c a 3f a 7c h 1f h

9 a c 3 a f 7 h c 1 h f

9 c a 3 f a 7 c h 1 f h

9 a c 3 a f 7 h c 1 h f

9 c a 3 f a 7 c h 1 f h

9c a 3f a 7c h 1f h

9a c 3a f 7h c 1h f

9. There are four sets of equivalent equations. Each column of the table above shows a set.10. 10,y x 10 ,x y 10 ,y x 10 ,x y 10 ,y x 10 ,y x 10x y

Solutions to Skills development 3

1. (a) Many values are possible.(b) Three possibilities are: , ,b a c a c b b c a . Yours should be checked by

substituting your values from part (a).2. (a) Many values are possible.

(b) Three possibilities are: , ,p q r p r q p q r . Yours should be checked bysubstituting your values from part (a).

3. (a) Many values are possible.(b) Three possibilities are: , ,x z y x y z y x z . Yours should be checked by

substituting your values from part (a).





1. 32. 83. (-4°)

4.2 1

or4 2

5. 106. 3.27. 10.61

8.5 1

or100 20

9. 14010. (0, 1)


1. (a) 11(b) 24(c) 36(d) (-2)(e) 23

2. (a) 9(b) 20(c) 21

(d) 5 1

14 4

(e) 16

3. (a) 3(b) 25(c) 83(d) 75(e) (-1)




2

2

12 4

(-3) 12 4 (-3)

9 12 (-12)

(-3) (-12) is false

x x

2

2

12 4

(-2) 12 4 (-2)

4 12 (-8)

(-8) (-8) is true

x x

2

2

12 4

(-1) 12 4 (-1)

1 12 (-4)

(-11) (-4) is false

x x

2

2

12 4

0 12 4 0

0 12 0

(-12) 0 is false

x x

2

2

12 4

1 12 4 1

1 12 4

(-11) 4 is false

x x

2

2

12 4

2 12 4 2

4 12 8

(-8) 8 is false

x x

2

2

12 4

3 12 4 3

9 12 12

(-3) 12 is false

x x

2

2

12 4

4 12 4 4

16 12 16

4 16 is false

x x

2

2

12 4

5 12 4 5

25 12 20

13 20 is false

x x

2

2

12 4

6 12 4 6

36 12 24

24 24 is true

x x

2

2

12 4

7 12 4 7

49 12 28

37 28 is false

x x

2

2

12 4

8 12 4 8

64 12 32

52 32 is false

x x

Solutions are 6 and (-2) as these values make the equation true.




1. (a) 8x (b) 11x (c) 10x (d) 5x (e) 18x (f) 9x (g) 3x (h) 4x (i) 4x (j) 5x (k) 9x (l) (-2)x



1. 4 should be circled2. 2453. (-6)

4.4

55. 86. 0.867. 238. 0.66…9. 0.0310. 100%


1. (a) − 9(b) + 3(c) ÷ 5(d) × 8(e) + 10(f) − 12(g) × 7(h) ÷ 6


1. You should end with your starting number.2. It should happen whatever number you started with.3. Each process is reversed within the sequence in an order that returns the number to where

it started.




1. (a) 11 + 4 gives 15 then 15 ÷ 3 = 5 ie x = 5(b) 17 − 2 gives 15 then 15 ÷ 5 = 3 ie x = 3(c) 0 + 8 gives 8 then 8 ÷ 4 = 2 ie x = 2(d) 17 − 12 gives 5 ie x = 5(e) 9 − 3 gives 6 then 6 ÷ 2 = 3 ie x = 3



1. 562. 3083. 7°

4.3

55. 106. 0.557. 68. 42%9. 010. Circle


1. (a) x = 3(b) x = 4

2. (a) x = (-1)(b) x = 2




1. Part (a) has a solution of x = 4 which substituted into all other parts makes them true andhence equivalent.

2. (a) and (e) have x = 8 as a solution.(b) and (g) have x = (-2) as a solution.(d) and (j) have x = 5 as a solution.

3. 4 × 3 + 7 = 19 is true hence the equation is equivalent.4. 4 12x 5. 3x 6. (a)

3 1 16


3 15


3 31 5

ie 5

x

x

x

x

x

x

(b) 5 6 19

5 6 6 19 6 6 is added to each side

5 25


5 51 5

ie 5

x

x

x

x

x

x




1. (a)7 1 15


7 14


7 71 2

ie 2

x

x

x

x

x

x

(b) 4 3 13


4 16


4 41 4

ie 4

x

x

x

x

x

x

(c) 3 9 0


3 9


3 31 3

ie 3

x

x

x

x

x

x

(d) 37 16 90


37 74


37 371 2

ie 2

x

x

x

x

x

x



7. Using equations


1. 1.052. 183. 6°

4.3 1

or6 2

5. 56. 8.77. 18. $129. Start at 15, then add 15 each time.10. 42°


1. Other equivalent expressions are possible.(a) 8x (b) 5x (c) 18x (d) 7y(e) 2( 11)x

(f) 2x x

(g) 5

x

(h) 4 6x 2. Other equivalent statements are possible.

(a) Eight more than a number(b) Six less than a number(c) The product of 9 and a number(d) The result of reducing a number by five is tripled.


1. Using x = 3 gives 2 + 3 + 3 = 5 + 3 which is true.2. 2 + x + x = 5 + x (any order is correct)3. Step 1: x + x = 3 + x

Step 2: x = 34. x = 3 agrees with the answer for question 1.




1. (a) Each avocado was $2 as shown here.3 4 10


3 6


3 31 2

ie 2

x

x

x

x

x

x

(b) The banknotes are $5 notes as shown here.5 3 28


5 25


5 51 5

ie 5

x

x

x

x

x

x

(c) The cost of each ticket is $13 as shown here.6 78


6 61 13

ie 13

x

x

x

x

(d) The string was originally 32 cm as shown here.12 20

12 12 20 12 12 is added to each side

ie 32

x

x

x



Solutions to Review tasks

Solutions to Task A

1. (a) Any three values can be given provided they fit the equation .x z y

For example, 5 − 2 = 3.

(b) Any three of the following equations would be correct.

, , , , , ,y x z z x y x z y x y z x y z z y x y z x

2. (a) 3x

(b) 7y

3. (a) 2x

(b) 2x

4. (a) 1x

(b) 3x

5. (a) 8 226.8 gives 28.35.x x So each coin weighs 28.35 g

(b) 4 3 10 gives 1.75.x x So each block of chocolate costs $1.75

Solutions to Task B

1. The original three diagrams plus diagrams 4 and 5 are shown here.

2. Diagram x 1 2 3 4 5 6 7

Squares y 1 5 9 13 17 21 25

3. 4 3y x is the correct equation for the values.



4. 4 3

4 30 3

120 3

ie 117

y x

y

5. 4 3 49

4 52

ie 13

x

x

x

6. It shows that the diagram with 49 squares is diagram 13.

MATHSAC021 SOLVING EQUATIONS ISBN: 9780730744412

australian curriculum mathematics year 7mathematics for the following year 7 content descriptions....

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