Title Cambridge Maths 7 Chapter 9 Algebra Fraction (Mathematics) Multiplication Variable (Mathematics) Equations 4.0 MB 48
##### Document Text Contents
Page 1

ChapterChapter Equations 1Equations 199
What you will learnWhat you will learn

Introduction to equationsIntroduction to equations

Solving equations by inspectionSolving equations by inspection

Equivalent equationsEquivalent equations

Solving equations systematicallySolving equations systematically

Equations with fractionsEquations with fractions

Equations with bracketsEquations with brackets

Formulas and relationshipsFormulas and relationships EXTENSIONEXTENSION

Using equations to solve problemsUsing equations to solve problems EXTENSIONEXTENSION

9A 9A

9B9B

9C9C

9D9D

9E9E

9F9F

9G9G

9H9H

ISBN: ISBN: 97811076269739781107626973 © © DDaavviid d GGrreeeennwwooood d eet t aall. . 22001133 CCaammbbrriiddgge e UUnniivveerrssiitty y PPrreessss

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Chapter 9Chapter 9 Equations 1Equations 1454566

Equations with fractionsEquations with fractions

Solving equations that involve fractions is straightforward once we recall that, in algebra,Solving equations that involve fractions is straightforward once we recall that, in algebra,
aa

bb
means means aa ÷÷ bb..

This means that if we have a fraction withThis means that if we have a fraction with bb on  on the denominatorthe denominator, we can , we can multiply both sides bymultiply both sides by bb to get a to get a

simpler, equivalent equation.simpler, equivalent equation.

Let’s start:Let’s start: Fractional differencesFractional differences
Consider these three equations.Consider these three equations.

aa
2 2 33

55
77

x  x  + +
== bb

22

55
3 3 77

x  x

+ + == cc 22
55

3 3 77
x  x  

  
 
   
+  + ==

•• Solve each of them (by inspection or systematically).Solve each of them (by inspection or systematically).

•• Why do two of the equations have the same solution?Why do two of the equations have the same solution?

9E9E

■■ Recall thatRecall that
aa

bb
means  means aa ÷÷ bb..

■■ The expressionsThe expressions
x  x

33
22++  and and

x  x  + +  2 2

33
are different, as are different, as demonstratdemonstrated in ed in these flow these flow charts.charts.

vsvs x  x
x  x

33

x  x

33
++ 2 2

÷÷33 ++22
x  x   x  x ++ 2 2

x  x ++ 22

33

++22 ÷÷33

■■ To solve an equation that has a fraction on one side, multiplyTo solve an equation that has a fraction on one side, multiply bothboth sides by the denominator. sides by the denominator.

■■ If neither side of an equation is a fraction, do not multiply by the denominator.If neither side of an equation is a fraction, do not multiply by the denominator.

■■ Sometimes it is wise to swap the LHS and RHS.Sometimes it is wise to swap the LHS and RHS.

For example:For example: 1212
33

11= = ++
x  x

becomes becomes
33

1 1 1212+ + == , which is easier to solve., which is easier to solve.

x  x

55
== 4 4

x  x == 20 20

××55 ××55

x  x

33
++ 5 5 == 8 8

. . . . ..

××33 ××33  ✗ ✗ Do not do thisDo not do this

x  x

33
++ 5 5 == 8 8

x  x

33
== 3 3

x  x == 9 9

−−55

××33

−−55

××33 ✓✓ Do this Do this

K   K
e   e
y   y
i    i   d   d
e   e
a   a
s   s

ISBN: ISBN: 97811076269739781107626973 © © DDaavviid d GGrreeeennwwooood d eet t aall. . 22001133 CCaammbbrriiddgge e UUnniivveerrssiitty y PPrreessss

Page 47

Chapter 9Chapter 9 Equations 1Equations 1474788

11 Classify each of the following equations as true or false.Classify each of the following equations as true or false.

aa 44 ++ 2 2 == 10 10 −− 2 2 bb 2(32(3 ++ 5) 5) == 4(1 4(1 ++ 3) 3) cc 55ww ++ 1 1 == 11, if 11, if ww == 2 2

dd 22 x  x ++ 5 5 == 12, if 12, if x  x == 4 4 ee  y y == 3 3 y y −− 2, if 2, if y y == 1 1 ff 44 == z z ++ 2, if 2, if z z == 3 3

22 Write an equation for each of the following situations. You do not need to solve the equations.Write an equation for each of the following situations. You do not need to solve the equations.

aa The sum of 2 andThe sum of 2 and uu is 22. is 22. bb The product ofThe product of k k  and 5 is 41. and 5 is 41.

cc WhenWhen z z is tripled the result is 36. is tripled the result is 36. dd The sum ofThe sum of aa and and bb is 15. is 15.

33 Solve the following equations by inspection.Solve the following equations by inspection.

aa  x x ++ 1 1 == 4 4 bb  x  x ++ 8 8 == 14 14 cc 99 ++ y y == 10 10

dd  y y −− 7 7 == 2 2 ee 55aa == 10 10 ff
aa

55
== 2 2

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

aa 22 x  x ++ 5 5 == 13  13 [[−− 5] 5] bb 77aa ++ 4 4 == 32  32 [[−− 4] 4]

cc 1212 == 3 3r r −− 3  3 [[++ 3] 3] dd 1515 == 8 8 p p −− 1  1 [[++ 1] 1]

55 Solve each of the Solve each of the following equations systematfollowing equations systematically and check your ically and check your solutions by substituting.solutions by substituting.

aa 55 x  x == 15 15 bb r r ++ 25 25 == 70 70 cc 1212 p p ++ 17 17 == 125 125 dd 1212 == 4 4bb −− 12 12

ee 55
33

22= = ++
x  x

ff 1313 == 2 2r r ++ 5 5 gg 1010 == 4 4qq ++ 2 2 hh 88uu ++ 2 2 == 66 66

66 Solve the following equations systematically.Solve the following equations systematically.

aa
33

44
66

uu

== bb
88

33
88

p p
== cc 33

2 2 11

33
==

++ x  x

dd
55

22
110 0 3300

y y
+ + == ee 44

2 2 2020

77
==

++ y y
ff

44

33
4 4 2424

x  x

+ + ==

77 Expand the brackets in each of the following expressions.Expand the brackets in each of the following expressions.

aa 2(32(3 ++ 2 2 p p)) bb 4(34(3 x  x ++ 12) 12) cc 7(7(aa ++ 5) 5) dd 9(29(2 x  x ++ 1) 1)

88 Solve each of these equations by expanding the brackets first. Check your Solve each of these equations by expanding the brackets first. Check your solutions by substituting.solutions by substituting.

aa 2(2( x x −− 3) 3) == 10 10 bb 2727 == 3( 3( x  x ++ 1) 1) cc 4848 == 8( 8( x  x −− 1) 1)

dd 6060 == 3 3 y y ++ 2( 2( y y ++ 5) 5) ee 7(27(2 z z ++ 1) 1) ++ 3 3 == 80 80 ff  2(5  2(5 ++ 3 3qq)) ++ 4 4qq == 40 40

99 Consider the equation 4(Consider the equation 4( x  x ++ 3) 3) ++ 7 7 x  x −− 9 9 == 10. 10.

aa IsIs x  x == 2 a solution? 2 a solution?

bb Show that the solution to this equation isShow that the solution to this equation is not not  a whole number. a whole number.

10 10 aa Does Does 3(23(2 x  x ++ 2) 2) −− 6 6 x  x ++ 4 4 == 15 have a solution? Justify your answer. 15 have a solution? Justify your answer.

bb State whether the following are solutions to 5(State whether the following are solutions to 5( x  x ++ 3) 3) −− 3( 3( x  x ++ 2) 2) == 2 2 x  x ++ 9. 9.

ii  x  x == 2 2 iiii  x  x == 3 3

1111 The formula for the area The formula for the area of a trapezium isof a trapezium is A A ==
11

22
hh((aa ++ bb), where), where hh is the  is the height of the trapezium, andheight of the trapezium, and aa

andand b b represent the parallel sides. represent the parallel sides.

aa Set up and solve an equation to find the area of a trapezium with height 20 cm and parallel sides ofSet up and solve an equation to find the area of a trapezium with height 20 cm and parallel sides of

15 cm and 30 cm.15 cm and 30 cm.

ISBN: ISBN: 97811076269739781107626973 © © DDaavviid d GGrreeeennwwooood d eet t aall. . 22001133 CCaammbbrriiddgge e UUnniivveerrssiitty y PPrreessss