Chapter 2 – Algebra I

ACTIVE MATHS 3 CH 02 SOLUTIONS 17

Chapter 2 – Algebra I

Exercise 2.1Q. 1. (i) 2(1) = 2

(ii) 2(−3)=−6

(iii) 3((1)+(−3))

=3(1−3)

=3(−2)

=−6

(iv) 2((1)−2(−3))+3(1)

=2(1+6)+3(1)

=2(7)+3(1)

=14+3

= 17

(v) (1)3

= 1

(vi) 4(1)2

= 4(1)

= 4

(vii) 3(−3)2

=3(9)

= 27

(viii) (1)(−3)

=−3

(ix) (1)2(−3)

=(1)(−3)

=−3

(x) (1)2(−3)2

=(1)(9)

=9

(xi) (1)(−3)3

=(1)(−27)

=−27

(xii) (1)2(−3)3

=(1)(−27)

=−27

Q. 2. (i) 3(2)+4(1)

=6+4

= 10

(ii) (2)+(1)+(−5)

=2+1−5

=−2

(iii) 2(2)+6(1)+4(−5)

=4+6−20

=−10

(iv) 4(2)−3(−5)−3(1)

=8+15−3

= 20

(v) 12(2)+2(−5)−8(1)

=24−10−8

=6

(vi) 3(1)−2(2)

=3−4

=−1

(vii) −6(2)+2(1)−4(−5)

=−12+2+20

= 10

(viii) −(2)−(1)−(−5)

=−2−1+5

= 2

Q. 3. (i) (4)(−2)(6)

=−48

(ii) (4)2+(−2)

=16−2

= 14

(iii) (4)2+3(−2)3−(6)2

=16+3(−8)−36

=16−24−36

=−44

ML7017_LCOL_02_AM3_Sol_v5.0.indd 17 03/04/17 8:11 pm

18 ACTIVE MATHS 3 CH 02 SOLUTIONS

(iv) (4)(−2)+(−2)(6)

=−8−12

=−20

(v) (4)2(−2)2(6)2

=(16)(4)(36)

=2,304

(vi) (4)2(−2)−(6)3

=(16)(−2)−216

=−32−216

=−248

Q. 4. (i) 2((−1)+(2))

=2(−1+2)

= 2(1)

= 2

(ii) 4((2)+(5)−(−1))

=4(2+5+1)

= 4(8)

=32

(iii) 5((2)+(−1))−2(5)

=5(2−1)−2(5)

=5(1)−2(5)

=5−10

=−5

(iv) 3((−1)−(5))2−(−1)(2)

=3(−1−5)2−(−1)(2)

=3(−6)2−(−1)(2)

=3(36)−(−1)(2)

=108+2

= 110

Q. 5. (i) Top=3(−1)+1

=−3+1

=−2

Bot=4(2)−(5)2

=4(2)−25

=8−25

=−17

∴ Ans = −2 ____ −17

= 2 ___ 17

(ii)

(−1) ____ 5 −(5)

___ 2 −(2)

=−1 __ 5−5 __ 2−

2 __ 1

= 2(−1)−5(5)−10(2)

__________________ 10

= −2−25−20 ____________ 10

=−47 ___ 10

(iii) 1 __ 5(5)+

1 __ 2(2)+2(2)

=1+1+4

=6

(iv) √ _________ (−1)+(2)

= √ ______ −1+2

= √ __ 1

= 1

(v) Top = (2)2+((5)+(−1))

=4+(5−1)

=4+4

= 8

Bot = 4(2)

= 8

Ans = √ __  8 __ 8 = √

__ 1 = 1

Q. 6. (i) 1 __ 3(3.14)(5)

2(10)

= 1 __ 3(3.14)(25)(10)

=2612 __ 3 cm3

(ii) 1 __ 3(3.14)(4)

2(5)

= 1 __ 3(3.14)(16)(5)

=8311 ___ 15 m3

(iii) 1 __ 3 ( 22 ___

 7 ) (56)2(200)

= 1 __ 3 ( 22 ___

 7 )(3136)(200) =657,0662 __

 3 mm3

Q. 7. (i) 2(2)(2)+2(2)(3)+2(2)(3)

=8+12+12

=32m2


19ACTIVE MATHS 3 CH 02 SOLUTIONS

(ii) 2(3.5)(12)+2(3.5)(9)+2(12)(9)

=84+63+216

=363cm2

(iii) 2(8)(9)+2(8)(12)+2(9)(12)

=144+192+216

=552m2

Q. 8. 100m hurdles 13.48sec

(i) p=9.2(27−13.48)1.8 ≈998.95

100mhurdles ≈998

(ii) p=1.8(198−75)1.3 ≈937.88

HighJump ≈937

(iii) p=56(11.68−1.5)1.05 ≈640.21

ShotPut ≈640

(iv) p=4.9(43−23.26)1.8 ≈1,051.53

200m ≈1,051

(v) p=0.19(651−210)1.4 ≈957.12

LongJump ≈957

(vi) p=16(36.36−3.8)1.04 ≈598.84

Javelin ≈598

(vii) p=0.1(254−130.47)1.9 ≈942.69

800m ≈942

2m10.47sec

=130.47sec

Total points =998+937+640+1,051+957+598+

942 =6,123

Exercise 2.2Q. 1. (i) 3a

(ii) 11b

(iii) −2b

(iii) −6d

(v) −4a

Q. 2. (i) e−5f

(ii) 5g+4h−7

(iii) 17j−2k

(iv) 8m−7n

(v) −p−2q

Q. 3. (i) 2x+3y+4+4x+2y+8

= 2x+4x+3y+2y+4+8

=6x+5y+12

(ii) 14x+5y−3−10x−3y+11

= 14x−10x+5y−3y−3+11

= 4x+2y+8

(iii) 2a+2b−2+5a−7b−5

= 2a+5a+2b−7b−2−5

= 7a−5b−7

(iv) −4p−2q+4−3p−2q−4

=−4p−3p−2q−2q+4−4

=−7p−4q

Q. 4. (i) 2x2+3x+4+x2+4x+1

= 2x2+x2+3x+4x+4+1

=3x2+7x+5

(ii) 5x2−5x+1−x2+2x−6

=5x2−x2−5x+2x+1−6

= 4x2−3x−5

(iii) 3a2−4a+9−4a2+4a−5

=3a2−4a2−4a+4a+9−5

=−a2+4

(iv) −p2−3p+1−3p2−2p−4

=−p2−3p2−3p−2p+1−4

=−4p2−5p−3

Exercise 2.3Q. 1. (i) 3ab

(ii) 12b

(iii) 10a2

(iv) 8c2

(v) −2bc

(vi) −d 2

(vii) 10e2

(viii) e2f



Q. 2. (i) x4

(ii) x3y2

(iii) 6x7

(iv) x8

(v) 6a10

(vi) −2y5

(vii) 20y5

(viii) 4b15

Q. 3. (i) a2b

(ii) a2b2

(iii) 6cd2

(iv) a3b

(v) 4x2y3

(vi) a3

(vii) −3y5

(viii) 10y2

Q. 4. (i) 8x3y3

(ii) −9x2y3

(iii) 50m2n2

(iv) −24a8

(v) 2x3y2

(vi) 10p4q4

(vii) −8p8q3

(viii) 10t7p4

Q. 5. (i) b2

(ii) 4b2

(iii) 9b2

(iv) −27b3

(v) 64a3b3

(vi) 4a 2b 2

(vii) 16x 4y 2

(viii) −8x 6y 3

Q. 6. (i) (2a)(b)+(a)(3b)

= 2ab+3ab

=5ab

(ii) (4p)(2q)−(2p)(q)+(4)(q)(p)

= 8pq−2pq+4pq

= 10pq

(iii) (3x)(x)(y)+(x)(xy)−(x2)(y)

=3x2y+x2y−x2y

=3x2y

(iv) (3ab)(−a)−(2a)(3b)+(4b)(a2)−(5)(a)(b)

=−3a2b−6ab+4a2b−5ab

= a2b−11ab

(v) (xy)2−3(xy)(x)−(2x2)(y)2

= x2y2−3x2y−2x2y2

=−x2y2−3x2y

Exercise 2.4Q. 1. (i) 3a+9

(ii) 3b−9

(iii) −4c+16

(iv) 5x−15y−5

Q. 2. (i) 3x+6+4x+4=7x+10

(ii) 2a+2+6a+2=8a+4

(iii) 2a−8−5a+5=−3a−3

(iv) 12b−6−3b+12=9b+6

Q. 3. (i) 2a2−2a+2+2a2+10a+8 = 4a2+8a+10

(ii) −12a2−4a+4−4a2+2a+4 =−16a2−2a+8

(iii) 5b2−10b−5+4b+2b2+2 = 7b2−6b−3

Q. 4. (i) 2x−y+6+(3x−y+1)−(x−2y−2)

= 2x−y+6+3x−y+1−x+2y+2

= 2x+3x−x−y−y+2y+6+1+2

= 4x+9

(ii) x2+3x+5+3(2x2−7x+20)− 2(x2−3x+4)

= x2+3x+5+6x2−21x+60− 2x2+60−8



= x2+6x2−2x2+3x−21x+6x+ 5+60−8

=5x2−12x+57

(iii) 3(2a2−3a−1)−(3a2−2a−3)− 4(a2+2a−1)

=6a2−9a−3−3a2+2a+3−4a2−8a+4

=6a2−3a2−4a2−9a+2a−8a−3+3+4

=−a2−15a+4

Q. 5. (i) x2+3x

(ii) y2−6y

(iii) 4a2+2a

(iv) 8b2−8b

(v) 2x3−2x

(vi) −5x(3x2−7)

−15x3+35x

Q. 6. (i) a2+a+a2+a = 2a2+2a

(ii) 6x2−14x−2x2+10x = 4x2−4x

(iii) 21a2−6a−12a2−4a=9a2−10a

(iv) 3x2−9x+3+5x2+15x = 8x2+6x+3

(v) 6y2−9y−2y2−6y = 4y2−15y

Exercise 2.5

Q. 1. 5x+2x _______ 10 = 7x ___

 10

Q. 2. y+2y

______ 8 = 3y

___ 8

Q. 3. 3a−5a _______ 15 =−2a ___

 15

Q. 4. 6c−c ______ 4 = 5c ___

 4

Q. 5. 14x+12x _________ 21 = 26x ____

 21

Q. 6. 3(2a+1)+4(a+5)

_________________ 12

= 6a+4a+3+20 _______________ 12

= 10a+23 ________ 12

Q. 7. 3(4a+1)+5(3a−1)

___________________ 15

= 12a+15a+3−5 ________________ 15

= 27a−2 _______ 15

Q. 8. 2(2x−3)+(x−2)

________________ 8

= 4x+x−6−2 ____________ 8

= 5x−8 ______ 8

Q. 9. 5x+3−3(x+2)

_______________ 9

= 5x−3x+3−6 _____________ 9

= 2x−3 ______ 9

Q. 10. 3(3y−1)−8(4y−5)

__________________ 24

= 9y−32y−3+40

________________ 24

= −23y+37

_________ 24

Q. 11. 4x−2 ______ 12 −

x−2 _____ 24 −

2 __ 3

= 2(4x−2)−1(x−2)−8(2)

______________________  24

= 8x−4−x+2−16 _________________ 24

= 7x−18 ________ 24

Q. 12. 1 __ 3 (x+1)−3 __ 2(3x−2)−5 __ 6 (x−3)

2(1)(x+1)−3(3)(3x−2)−1(5)(x−3)

_________________________________  6

= 2(x+1)−9(3x−2)−5(x−4)

__________________________  6

= 2x+2−27x+18−5x+15 ________________________  6

= −30x+35 _________ 6

Q. 13. 9(5x+1)−10(2x+3)+46

________________________  15

= 45x−20x+9−30+46 _____________________  15

= 25x+25 ________ 15

= 5x+5 ______ 3



Exercise 2.6Q. 1. (i) x2+5x+6

(ii) x2+5x+4

(iii) x2−4x−21

(iv) x2+4x−5

(v) y2−12y+32

Q. 2. (i) 8x2+4x+2x+1

= 8x2+6x+1

(ii) 6y2+2y−12y−4

=6y2−10y−4

(iii) 15x2−6x−15x+6

=15x2−21x+6

(iv) 12y2−24y−6y+12

= 12y2−30y+12

(v) 4b−2−4b2+2b

=−4b2+6b−2

Q. 3. (i) (x+y)(x+y)

= x(x+y)+y(x+y)

= x2+xy+xy+y2

= x2+2xy+y2

(ii) (p−q)(p+q)

= p(p+q)−q(p+q)

= p 2 +pq−pq−q2

= p 2−q2

(iii) (m−n)(m−n)

= m (m−n)−n(m−n)

= m2−mn−mn+n2

= m2−2mn+n2

(iv) (2h−w)(2h−3w)

= 2h(2h−3w)−w(2h−3w)

= 4h2−6hw−2hw+3w2

= 4h2−8hw+3w2

Q. 4. (i) x2+2x+1 (ii) x2+6x+9

(iii) x2−4x+4 (iv) x2−10x+25

(v) 4y2+4y+1

Q. 5. (i) x3+x2+x+x2+x+1

= x3+2x2+2x+1

(ii) 2x3+x2−x−6x2−3x+3

= 2x3−5x2−4x+3

(iii) 16x3−8x2−12x+4x2−2x−3

=16x3−4x2−14x−3

Q. 6. (i) x(x+3)(x+4)

= x(x2+3x+4x+12)

= x(x2+7x+12)

= x3+7x2+12x

(ii) h(10−2h)(10−2h)

= h(100−20h−20h+4h2)

= h(100−40h+4h2)

= 100h−40h2+4h3

(iii) a(a−b)(a+b)

= a(a2+ab−ab−b2)

= a(a2−b2)

= a3−ab2

Q. 7. (i) (2x+3)3

= (2x+3)(2x+3)(2x+3)

= (2x+3)(4x2+6x+6x+9)

= (2x+3)(4x2+12x+9)

= 8x3+24x2+18x+12x2+36x+27

= 8x3+36x2+54x+27

(ii) y 3−6y 2+12y−8

Q. 8. (i)

2x +3

3x 6x2 +9x

−5 −10x −15

Area=6x2+9x−10x−15

=6x2−x−15 (ii)

x2 −5x +25

x x3 −5x2 +25x

+5 −5x2 −25x +125

Area = x3−5x2+5x2+25x−25x+125

= x3+125



Exercise 2.7Q. 1. x(x+5)

Q. 2. x(x−7)

Q. 3. x(x−2)

Q. 4. x(x−15)

Q. 5. a(a−11)

Q. 6. x(x−14)

Q. 7. ad+ae+cd+ce

= a(d+e)+c(d+e)

= (a+c) (d+e)

Q. 8. ap+aq+bp+bq

= a( p+q)+b( p+q)

= (a+b) (p+q)

Q. 9. mp – mq+np – nq

= m(p – q)+n(p – q)

= (m+n) (p – q)

Q. 10. a(c −d)−b(c −d)

(c −d ) (a −b)

Q. 11. xy – y – 4x+4

= y(x – 1) – 4(x – 1)

= (x – 1) (y – 4)

Q. 12. 3p(2a – e)+a(2d – e)

Q. 13. r(2x −s)−5y(2x −s)

(2x −s)(r −5y)

Q. 14. pm+np+nq+mq

= pm+np+mq+nq

= p(m+n)+q(m+n)

= (p+q) (m+n)

Q. 15. (x−4)(x+4)

Q. 16. (x−3)(x+3)

Q. 17. (x−10)(x+10)

Q. 18. (x−2)(x+2)

Q. 19. (b−5)(b+5)

Q. 20. (y−11)(y+11)

Q. 21. (x−15)(x+15)

Q. 22. (x+3)2

Q. 23. (x+7)(x+1)

Q. 24. (x+9)(x+4)

Q. 25. (x+2)2

Q. 26. (x−3)(x−9)

Q. 27. (x+7)(x−2)

Q. 28. (x−7)(x+2)

Q. 29. (x−4)(x−8)

Q. 30. (x−8)2

Q. 31. (x−4)(x+1)

Q. 32. (x+9)(x−2)

Q. 33. (x−9)(x+7)

Q. 34. (x−6)(x+4)

Q. 35. (x−7)(x−8)



Q. 36. (b)2–(19)2

(b −19)(b +19)

Q. 37. pc + bd – bc − dp

= pc − bc − dp + bd

= c( p − b) − d( p − b)

=( p − b) (c − d )

Q. 38. (x+10)(x−7)

Q. 39. x2−10x −39

(x−13)(x+3)

Q. 40. (x+7)(x+8)

Q. 41. (x−10)(x+9)

Q. 42. (3−x)(3+x)

Exercise 2.8Q. 1. 2x(x+3)

Q. 2. 3x(x+7)

Q. 3. 5x(x−5)

Q. 4. 4x(x−4)

Q. 5. 3a(a −13)

Q. 6. b(2b+3)

Q. 7. a(2a–15)

Q. 8. x(5x+2)

Q. 9. 3x2+7x

x(3x+7)

Q. 10. p(11p –3)

Q. 11. 4(x2−4)

= 4(x−2)(x+2)

Q. 12. (3b)2 – (2)2

(3b+2)(3b – 2)

Q. 13. (8x)2−(7)2 = (8x−7)(8x+7)

Q. 14. 25(a2−4)=25(a−2)(a+2)

Q. 15. (7b)2–(6)2

(7b+6)(7b–6)

Q. 16. (11x)2 −(5)2 = (11x−5)(11x+5)

Q. 17. (9y−14)(9y+14)

Q. 18. (13x)2 – (18)2

(13x+18)(13x – 18)

Q. 19. (2x+3)(x+1)

Q. 20. (5x+1)(x+1)

Q. 21. (7p+1)(p+2)

Q. 22. (5x+2)(x+2)

Q. 23. (3q+2)(q+2)

Q. 24. (3x+2)(x+5)

Q. 25. (7x−5)(x+1)

Q. 26. (3y−5)(y−1)

Q. 27. (3x−7)(x+1)

Q. 28. (2x−5)(x+2)

Q. 29. (7a−5)(a+1)

Q. 30. (5x−3)(x+4)



Q. 31. (2b−3)(b−5)

Q. 32. (5x−3)(x−3)

Q. 33. (5p−7)(p+7)

Q. 34. (11x–5)(x+6)

Q. 35. (4x−5)(x+1)

Q. 36. (8q+1)(q+2)

Q. 37. (4x−7)(x+3)

Q. 38. 4(225x2−49)=4(15x−7)(15x+7)

Q. 39. 2(3x2+5x+2)=2(3x+2)(x+1)

Q. 40. 3(x2−6x+8)=3(x−4)(x−2)

Exercise 2.9Q. 1. a9

Q. 2. 8a

Q. 3. 5a2b

Q. 4. 5a5b

Q. 5. −3x4

Q. 6. 2xy

Q. 7. 7x4y2

Q. 8. 8xy 3z2

Q. 9. 5a+50 _______ 5 = 5(a+10)

________ 5 = a+10

Q. 10. 4b+2 ______ 2 =

2(b+1) _______ 2 = b+1

Q. 11. 2p−8

______ 2 =

2(p−4) _______ 2 = p−4

Q. 12. 10x−5 _______ −5 = 5(2x−1)

________ −5 =−(2x−1)=−2x+1

Q. 13. 2x+9 ______ 2x+9 = 1 __

 1 = 1

Q. 14. 3x+1 _______ 15x+5 = 3x+1 ________ 

5(3x+1) = 1 __

 5

Q. 15. 2x 2+10x _________ x+5 =

2x(x+5) ________ x+5 = 2x

Q. 16. 7x 2−14x _________ x−2 =

7x(x−2) ________ x−2 = 7x

Q. 17. 2x 2+4x−6 ___________

 2 = 2(x 2+2x−3)

____________ 2 = x2+2x−3

Q. 18. x(x−4)

_______ x−4 = x

Q. 19. (x+2)(x+5)

____________ x+2 = x+5

Q. 20. (x−8)(x−6)

____________ x−8 = x−6

Q. 21. 2x+3 2x+14x 2+8x+3 – (4 x 2+2x) 6x+3 –(6x+3) 0 ∴ Ans = 2x+3

Q. 22. x −2 2x–92x2−13x+18 – (2x2−9x) −4x+18 −(−4x+18) 0 ∴ Ans = x − 2

Q. 23. 2x−5 4x – 7 8x2−34x+35 – (8x2−14x) –20x+35 −(−20x+35) 0 ∴ Ans = 2x − 5

1

1



Q. 24. 3 _____________  

3x2−18x+24 = 3 ____________

  3(x2−6x+8)

= 1 __________ 

x2−6x+8

Q. 25. x−8 _______ 

x2−64 = x−8 ____________

  (x−8)(x+8)

= 1 _____ x+8

Q. 26. x+4 __________ 

x2+6x+8 = x+4 ____________

  (x+4)(x+2)

= 1 _____ x+2

Q. 27. 2x+9 _____________  

2x2+15x+27 =

(2x+9) _____________  

(2x+9)(x+3) = 1 _____

 x+3

Q. 28. 3x+5 ____________  

9x2+18x+5 = 3x+5 ______________

  (3x+1)(3x+5)

= 1 ______ 3x+1

Q. 29. 8x+5 ______________  

16x2−14x−15 = 8x+5 ______________

  (8x+5)(2x−3)

= 1 ______ 2x−3

Q. 30. 5x+9 ______________  

60x2+83x−45 = 5x+9 _______________

  (5x+9)(12x−5)

= 1 _______ 12x−5

Q. 31. 5x +13

5x–1325x2+0x–169 –(25x2–65x) 65x–169 –(65x–169) 0 ∴ Ans = 5x+13

Q. 32. g(x) = 11x2+61x–30 _______________ 11x–5

= (x+6)(11x –5)

_______________ 11x –5

g(x) = x +6

Q. 33. h(x) = −6x2 + 11x – 4 _____________ 3x – 4

–2x +1 3x–4–6x2+11x – 4 –(–6x2+8x) 3x – 4 –(3x – 4) 0 ∴ h(x) = –2x+1

Q. 34. f(x) = 10x2+61x+91 ______________ −2x−7

−5x −13 −2x−710x2+61x+91 −(10x2+35x) 26x+91 −(26x+91) 0 ∴ f(x) = −5x −13

Exercise 2.10

Q. 1. 1 _____ x +2+

1 __ x =

1(x)+1(x+2) _____________ 

x(x+2)

= x+x+2 ________ 

x(x+2)

= 2x+2 _______ 

x(x+2)

Q. 2. 2 __ x + 1 _____

 x −2 = 2(x−2)+x

__________ 

x(x−2)

= 2x−4+x _________ 

x(x−2)

= 3x−4 _______ 

x(x−2)

Q. 3. 11 ______ 2x −1−3 __ x =

11(x)−3(2x−1) _______________  

x(2x−1)

= 11x−6x+3 ___________ 

x(2x−1)

= 5x+3 ________ 

x(2x−1)

Q. 4. 2 _____ x −3+

4 _____ x +2 =

2(x+2)+4(x−3) ________________  

(x−3)(x+2)

= 2x+4+4x−12 _______________  

(x−3)(x+2)

= 6x−8 ____________  

(x−3)(x+2)

Q. 5. 5 ______ 2x +1−

3 _____ x +4 = 5(x+4)−3(2x+1)

_________________  

(2x+1)(x+4)

= 5x+20−6x −3 _______________  

(2x+1)(x+4)

= 17−x _____________  

(2x+1)(x+4)

Q. 6. 2 _____ x−8−

3 _____ x+4 =

2(x+4)−3(x−8) ________________  

(x−8)(x+4)

= 2x+8−3x+24 _______________  

(x−8)(x+4)

= −x+32 ____________  

(x−8)(x+4)

OR ( 32−x ____________  

(x−8)(x+4) )



Q. 7. 1 ______ 3x −1−

4 ______ 2x −3 =

1(2x−3)−4(3x−1) __________________  

(3x−1)(2x−3)

= 2x−3−12x+4 _______________  

(3x−1)(2x−3)

= 1−10x ______________  

(3x−1)(2x−3)

Q. 8. 7 ______ 2x −1−

3 ______ 1 +2x = 7(1+2x)−3(2x−1)

__________________  

(2x−1)(1+2x)

= 7+14x−6x+3 _______________  

(2x−1)(1+2x)

= 8x+10 ______________  

(2x−1)(1+2x)

Q. 9. 8 ______ 2x +5−

1 __ 3 = 8(3)−1(2x+5)

______________  

3(2x+5)

= 24−2x−5 __________ 

3(2x+5)

= 19−2x ________ 

3(2x+5)

Q. 10. 4 __ x +2 ___

 3x = 4(3)+2(1)

__________ 3x

= 12+2 ______ 3x

= 14 ___ 3x

Q. 11. 5 ___ 2x −1 ___

 4x = 5(2)−1(1)

__________ 4x

= 10−1 ______ 4x

= 9 ___ 4x

Q. 12. 5 ______ 4x−3+

4 ______ 3−4x = 5 ______

 4x−3−4 ______ 4x−3

= 1 ______ 4x−3

Exercise 2.11Q. 1. x=3

Q. 2. y = 2

Q. 3. 2x = 8

x = 4

Q. 4. 3a = 27

a=9

Q. 5. 3x=−3

x=−1

Q. 6. 7x=−21

x=−3

Q. 7. 5y = 20

y = 4

Q. 8. x = 7

Q. 9. −t=−3

t=3

Q. 10. 6x=5

x = 5 __ 6

Q. 11. 4x=−8

x=−2

Q. 12. 2x=−5

x=−5 __ 2

Q. 13. 2x−2=4x

−2x = 2

x=−1

Q. 14. 4x+12=3x+6

x=−6

Q. 15. 10y−5=2y−2

8y=3

y = 3 __ 8

Q. 16. 3x+3=2x−6+7

x=−2

Q. 17. 4x+2−3x+3=9

x = 4



Q. 18. 35x+15=6−2x−3+x

35x+15=3−x

36x=−12

x=−1 __ 3

Q. 19. 4x−8=12−6x+3

10x=23

x=2.3

Q. 20. 3x−18+15=4x−4+4

−x=3

x=−3

Q. 21. −12x+6=6x−2+18

−18x = 10

x=−5 __ 9

Q. 22. 11a+77=4−10a+10

21a=−63

a=−3

Q. 23. 10x−5=4x−6+4x−23

2x=−24

x=−12

Q. 24. 3x−6=7x+35−13

−4x = 28

x=−7

Q. 25. 12x−18+25x+50=x−4

36x=−36

x=−1

Q. 26. 11 = 7x+7−6+16x−3x

11 = 20x+1

10 = 20x

1 __ 2 = x

x = 1 __ 2

Q. 27. 3y−3=1+4y

−4=y

y=−4

3(−4−1)=1+4(−4)

3(−5)=1−16

−15=−15

True

∴ y=−4issolution

Q. 28. 11a−2=5a−10−2a

11a−2=3a−10

8a=−8

a=−1

11(−1)−2=5(−1−2)−2(−1)

−11−2=5(−3)−2(−1)

−13=−15+2

−13=−13

True

∴ a=−1issolution

Q. 29. 2b+6−4=−b−2−3

3b=−7

b=−7 __ 3

2 ( −7 ___ 3+3)−4=−( −7 ___

 3+2)−3 2 ( 2 __

 3 )−4=−( −1 ___ 3 )−3

4 __ 3−4=

1 __ 3−3

−8 __ 3=−

8 __ 3

True

∴ b=−7 __ 3 is solution

Exercise 2.12Q. 1. x+3=5

x = 2

Q. 2. x+4=16

x = 12



Q. 3. 2(x+1)=3x

2x+2=3x

2 = x

x = 2

Q. 4. 9x−4=8

9x = 12

x = 12 ___ 9

x = 4 __ 3

Q. 5. x−3x=30

−2x=30

x=−15

Q. 6. 3(x+1)=4(x−1)

3x+3=4x−4

7 = x

x = 7

Q. 7. 2x−1=3x−12

11 = x

x = 11

Q. 8. 3(5y−1)+8(2y−4)=24(5)

15y−3+16y−32=120

31y−35=120

31y=155

y=5

Q. 9. 3x−2−4(x−1)=0

3x−2−4x+4=0

−x+2=0

x = 2

Q. 10. 3(2x+2)−7(4x−1)=21(−3)

6x+6−28x+7=−63

−22x+13=−63

−22x=−76

x = 38 ___ 11

Q. 11. 2(t−3)+2t+3=4(3)

2t−6+2t+3=12

4t=15

t = 15 ___ 4

Q. 12. 4(x−2)+5(x−3)=2(11)

4x−8+5x−15=22

9x−23=22

9x=45

x=5

Q. 13. 4(x+1)+3(2x−9)=17

4x+4+6x−27=17

10x−23=17

10x = 40

x = 4

Q. 14. 7(4x+5)−5(2x−3)=32

28x+35−10x+15=32

18x+50=32

18x=−18

x=−1

Q. 15. 3(x−4)−2(x−3)=30

3x−2x−12+6=30

x−6=30

x=36

Q. 16. 2(−x+2)+5x+2=6(1)

−2x+4+5x+2=6

3x+6=6

3x = 0

x = 0

Exercise 2.13Q. 1. (i) x=3

(ii) y = 4

(iii) 4a = 4

a = 1



(iv) 2b = 14

b = 7

(v) 4y=16−x

y = 16−x ______ 4

(vii) 2b−5=3c

2b−5 ______ 3 = c

(vii) 2r=−4t

r=−2t

(viii) −4a+3=2a−1

−4a−2a=−1−3

−6a=−4

−a = −4 ___ −6 = 2 __

 3

Q. 2. A __ l = w

Q. 3. 2K ___ 

v2 = m

Q. 4. ± √ __  E __ m = c

Q. 5. b−a _____ 2 = c

Q. 6. Fr __ 

v2 = m

Q. 7. 3a−b = c

Q. 8. v2 = 2gh

v2 ___ 2h = g

Q. 9. 2s−a−c = b

Q. 10. s−ut = 1 __ 2 at2

2s−2ut = at2

2s ___ 

t2−2u ___

 t = a or a = 2s−2ut ________ 

t2

Q. 11. 3x+y=6z

y=6z−3x

Q. 12. b(a+c) = c

b = c _____ a+c

Q. 13. 1 __ a +1 __

 b = 1 __ c

bc+ac = ab

ac = ab−bc

ac = b(a−c)

ac _____ a−c = b

Q. 14. pq−qr = c

q(p−r) = c

q = c ____ p−r

Q. 15. b−bx = x−a

b(1−x) = x−a

b = x−a _____ 1−x

Q. 16. T ___ 2p = √ 

__  l __ g

T 2 ____ 

4p 2 = l __ g

T 2g

____ 

4p 2 = l

Q. 17. a = bc _____ b+c

ab+ac = bc

ac = bc−ab

ac = b(c−a)

ac _____ c−a = b

Q. 18. (i) V = M __ D

(ii) 50 ___ 1.2 = 41 2 __

 3 cm3

Q. 19. (i) 3x−y

______  a+b = k

3x−y = k(a+b)

3x−k(a+b) = y

(ii) 3x−y = k(a+b)

3x = k(a+b)+y

x = k(a+b)+y

__________ 3

Q. 20. (i) V=3.14(5)2(20)=1,570cm3

(ii) V ___ 

pr2 = h

h = 3,768

_________ 

3.14(10)2 = 12 cm



(iii) V ___ ph = r2

√ ___

  V ___ ph = r (since r > 0)

      ∴ r = √ ______

  3,773

______  22 ___ 7 × 2

= √ ______

  3,773

______  44 ___ 7

= √ _______ 600.25

= 24.5 cm

Q. 21. (i) T = 2p √ _____

  2 _____ 9.81 = 2(3.14) √ _____

  2 _____ 9.81

= 2.84 seconds

(ii) T = 2p √ __  l __ g

T ___ 2p = √ 

__  l __ g

T 2 ____ 

4p 2 = l __ g

T 2g

____ 

4p 2 = l

(3)2(9.81)

_________ 

4(3.14)2 = 2.24 metres

Exercise 2.14Q. 1. 2x = 8

x = 4

4 + y = 7

y = 3

∴ x = 4, y = 3

Q. 2. 2x = 16

x = 8

8 + y = 13

y = 5

∴ x = 8, y = 5

Q. 3. 2a = 14

a = 7

7 + b = 4

b=−3

∴ a = 7, b=−3

Q. 4. 4x + 3y = 10

−(3x + 3y = 9)

x = 1

1+y = 3

y = 2

∴ x = 1, y = 2

Q. 5. p + 2q = 11

−(4p + 2q = 20)

−3p=−9

p = 3

2(3) + q = 10

q = 4

∴ p = 3, q = 4

Q. 6. −x=−2

x = 2

2(2) + y = 9

y = 5

∴ x = 2, y = 5

Q. 7. 4a + 10b=−30

−(4a + 3b=−9)

7b=−21

b=−3

2a+5(−3)=−15

2a = 0

a = 0

∴ a = 0, b=−3

Q. 8. 5x=−5

x=−1

3(−1)+5y = 17

5y = 20

y = 4

∴ x=−1,y = 4

ML7017_LCOL_02_AM3_Sol_v5.0.indd 31 08/04/17 11:07 AM


Q. 9. 6x+21y = 81

−(6x+10y=26)

11y=55

y=5

2x+7(5)=27

2x=−8

x=−4

∴ x=−4,y=5

Q. 10. 4p+3q=−21

−(4p+18q=−66)

−15q=45

q=−3

2p+9(−3)=−33

2p−27=−33

2p=−6

p=−3

∴ p=−3,q=−3

Q. 11. 2x−5y = 0

−(2x+8y=26)

−13y=−26

y = 2

x+4(2)=13

x=5

∴ x=5,y = 2

Q. 12. e−3f−5=0 ⇒ e−3f=5

5e−f+17=0×3⇒ −(15e−3f=−51)

−14e=56

e=−4

−4−3f=5

−3f=9

f=−3

Q. 13. 5x−5y=−5

4x+5y=−13

9x=−18

x=−2

−2−y+1=0

−y−1=0

∴ y=−1

∴ x=−2,y=−1

Q. 14. x+3y=−11

2x+y=−12

2x+6y=−22

−(2x+y=−12)

5y=−10

y=−2

x+3(−2)=−11

x−6=−11

x=−5

∴ x=−5,y=−2

Q. 15. 5x−3y = 12

3x+2y = 11

10x−6y = 24

9x+6y=33

19x=57

x=3

3(3)+2y = 11

2y = 2

y = 1

∴ x=3,y = 1

Q. 16. x+2y=26

3x−7y = 0

3x+6y = 78

−(3x−7y = 0)

13y = 78

y=6

x+2(6)=26

x+12=26

x = 14

∴ x = 14, y=6



Exercise 2.15

Q. 1. (i) x=−2 __ 3

y = 3x + 2

–1

10

0

2

3

4

5

–2

–3

–4

0.33 x

y

–0.67, 0

–0.67 –0.33

(ii) x=3

–2 –1–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

–11

–12

00

1

2

3

4

5

1 2 3 4(3,0)

5 6 7 x

y

y = 4x – 12

(iii) x=−5

–7 –6 –5 –4 –3 –2

(–5,–5)

–1–1

–2

–3

–4

–5

–6

–7

001

2

3

4

5

6

7

8

9

10

y

1 x

y = 3x + 10

y = –5

(iv) x = 2

–4 –3 –2 –1–1

1

2

3

4(2,5)5

6y

0

–2

–3

0 1 2 3 4 5 x

y = 2

x + 1

y = x + 3

(v) x=−7

–11–10–9 –8 –7 –6 –5 –4 –3 –2–2

(–7,10)

0

2

4

6

8

10

12

–1 0 1 2 3 4 x

yy = –(x – 3)

y = 10

Q. 2. (i) 2x+y=6

Let x = 0 Let y = 0

0+y=6 2x+0=6

y=6 x=3

(0,6) (3,0)

2x−y = 2

Let x = 0 Let y = 0

2(0)−y = 2 2x−0=2

−y = 2 2x = 2

y=−2 x = 1

(0,−2) (1,0)

–3 –2 –1–1

1

2

3

4

(2,2)

5

6y

0

–2

–3

0 1 2 3 4 5 6 7 8 x

y = 2x – 2

y = –2x + 6

x = 2, y = 2



(ii) 2x+y = 4 4x−5y = 20

y=−2x+4 −5y=−4x+20

y = 4 __ 5 x−4

m=−2 m = 4 __ 5

y intercept (0,4) yintercept(0,−4)

–3 –2 –1–1

1

2

3

4

5

6y

0

–2

–3

0 1 2 3 4 5 6 7 8 x

y = –2x + 4

y = x – 44–5

, 2 67 –15

7

x = 2 6 __ 7 , y=−15 __

 7

(iii) x+2y=5 3x−y=−6

2y=−x+5 −y=−3x−6

y=−1 __ 2 x+5 __

 2 y=3x+6

m=−1 __ 2 m=3

y intercept (0, 5 __ 2 ) yintercept(0,6)

–3 –2 –1–1

1

2

3

4

(–1,3)

5

6y

0

–2

–3

0 1 2 3 4 5 6 7 8 x

y = 3x + 6

y = – x + 5–21–2

x=−1,y=3

(iv) x−y = 4 x+5y=−2

−y=−x+4 5y=−x−2

y = x−4 y=−1 __ 5 x−2 __

 5

m = 1 m=−1 __ 5

y-intercept(0,−4) y-intercept ( 0,−2 __ 5 )

–4 –3 –2 –1–1

1

2

3

4

(3,–1)

y

0

–2

–3

–4

–5

0 1 2 3 4 5 6 7 x

y = x – 4

y = – x – 2–51–5

x=3,y=−1

Q. 3. (a) (i) x = 2, y = 2

–4 –3 –2 –1–1

1

2

3

4

(2,2)

y

0

–2

–3

–4

–5

0 1 2 3 4 5 6 7 x

y = 2

x – 2

y = x + 2–56–5

(ii) x = 1, y = 1 __ 2

–4 –3 –2 –1–1

1234

(1,0.5)

y

0

–2–3–4–5

0 1 2 3 4 5 6 7 x

y = 3x – 5–2

y = – x + 11–2

(b) (i) 2(2)−(2)=2

4−2=2

2 = 2

True

2(2)−5(2)=−6

4−10=−6

−6=−6

True

      ∴ (2,2) is solution



(ii) 1+2( 1 __ 2 ) = 2

1+1=2

2 = 2

True

6(1)−2( 1 __ 2 )=5

6−1=5

5=5

True

      ∴ ( 1, 1 __ 2 ) is solution

Exercise 2.16Q. 1 (i) 8,9,10

(ii) 6,7,8

(iii) −8,−7,−6

(iv) 1,2,3

(v) −4,−3,−2

Q. 2 (i) All natural numbers that are greater than 4

(ii) Allintegersthatarelessthan−1

(iii) All real numbers less than or equal to 12

(iv) All natural numbers less than 2

(v) Allrealnumbersgreaterthanorequalto3

Q. 3 (i) x 9,x ∈ N

(ii) x –1, x ∈ Z

(iii) x 5,x ∈ R

(iv) x 3,x ∈ N or x 2, x ∈ N

(v) x −2,x ∈ Z or x −3,x ∈ Z

(vi) x 2, x ∈ R

(vii) x 4, x ∈ Z or x 3,x ∈ Z

(viii) x 19,x ∈ R

Q. 4 (i) x 5orx 6,x ∈ N

(ii) x −6orx −7,x ∈ Z

(iii) x 5,x ∈ R

(iv) x −3,x ∈ R

(v) x 0, x ∈ R

Q. 5. (i) –2 –1 0 1 2 3 4 5 6 7

(ii) –2 –1 0 1 2 3 4 5 6 7

(iii) –2 –1–4–5 –3 0 1 2 3 4

(iv) –2 –1–4–5 –3 0 1 2 3 4

(v) –2 –1–4–5 –3 0 1 2 3 4

(vi) –2 –1–4–5 –3 0 1 2 3 4

(vii) –2 –1–4–5 –3 0 1 2 3 4

(viii) –2 –1–4–5–6–7 –3 0 1 2

(ix) –2 –1–3 0 1 2 3 4 5 6

(x) –2 –1–4–5 –3 0 1 2 3 4

Exercise 2.17Q. 1. x 2, x ∈ N

0 1 2 3 4

Q. 2. 4x−4 8, x ∈ N

4x 12

x 3

2 3 4 5

Q. 3. 3x+39,x ∈ N

3x 6

x 2

0 1 2

Q. 4. 5x −5,x ∈ Z

x −1

–3–4 –2 –1 0 1

Q. 5. 7x −7,x ∈ R

x −1

–3–4 –2 –1 0 1

Q. 6. 3x 12, x ∈ R

x 4

3 4 5 6 7

Q. 7. x−5−21,x ∈ R

x −16

–17 –16 –15 –14 –13



Q. 8. 5x 10, x ∈ Z

x 2

–2 –1 0 1 2 3

Q. 9. x −7,x ∈ Z

–8 –7 –6 –5 –4

Q. 10. −x 1, x ∈ R

x −1

–2 –1 0 1 2 3

Q. 11. x+1 10, x ∈ R

x 9

5 6 7 8 9 10

Q. 12. −2x −7,x ∈ N

2x 7

x 7 __ 2

0 1 2 3 4 5

Q. 13. 2(5x−2) 8x+10,x ∈ R

10x−4 8x+10

2x 14

x 7

5 6 7 8 9

Q. 14. 2x+45x−2,x ∈ R

−3x −6

3x 6

x 2 10–1 2 3 4

Q. 15. 6x+33x−9,x ∈ R

3x −12

x −4

–5 –4 –3 –2 –1 0

Q. 16. −9(2x+3)−8(x+4),x ∈ Z

−18x−27−8x−32

−10x −5

10x 5

x 1 __ 2

–1 0 1 2 3 4

Q. 17. 4x−12−2x−2+2,x ∈ R

6x 12

x 2

–1 0 1 2 3 4

Q. 18. 12x+4+2x−6+9 0, x ∈ Z

14x+7 0

14x −7

x −1 __ 2

–2 –1 0 1 2 3

Exercise 2.18Q. 1. (i) 5(7)=35

(ii) 7(7)=49

(iii) 7n

Q. 2. (i) 60

(ii) 4(60)=240

(iii) 60t

Q. 3. (i) 3(3)+10=19

(ii) 3x+y

Q. 4. (i) 22+4=26

(ii) x+4

Q. 5. (i) 45l

(ii) 3 __ 4 m

Q. 6. x+n

Q. 7. (i) 5,6,7,8,9

(ii) n+1,n+2,n+3,n+4,n+5



Q. 8. (i) 12,14,16,18

(ii) p+2,p+4,p+6,p+8

Q. 9. (i) 33,35,37

(ii) q+2,q+4,q+6

Q. 10. (i) 40−3=37

(ii) x __ 2−3

Q. 11. (i) 100−50=€50

(ii) 100−70=€30

(iii) €(100−x)

Q. 12. (i) 29−17=12

(ii) 29−x

Q. 13. 2(10x)+2(4x−1)

= 20x+8x−2

= (28x−2)m

Q. 14. € ( 1000−x ________ 3 ) for the three youngest

€x for the eldest

Q. 15. C(€)=90t+60

Q. 16. h(cm)=12+0.5d h = height of plant (cm) d = day

Q. 17. 12 units

Option 1 Option 2

100+5(12) 15(12) €160 €180 Option 1 is better if visiting gym 12 times in a

year.

Q. 18. (i) t(m) = 40x+20

(ii) t(m)=50x+20

(iii) t(m)=60x+30

Q. 19. 253+11x+x−50

S(€)=(12x+203)

Q. 20. (i) 0.25

(ii) Y(€)=30,000+0.25x

Exercise 2.19Q. 1. (i) 2x+17

(ii) 2x+17=35

(iii) 2x+17=35

2x = 18

x=9

Q. 2. (i) 3y−7

(ii) 3y−7=26

(iii) 3y−7=26

3y=33

y = 11

Q. 3. (i) 8x+60=180

(ii) y−10+2y+5+2y−15=180

5y−20=180

(iii) A 8x=180−60 B 5y=180+20

8x=120 5y = 200

x=15 y = 40

∴60°,45°,75° ∴65°,30°,85°

Q. 4. (i) 4(x+2)

(ii) 6(x−1)

(iii) 4(x+2)=6(x−1)

(iv) 4x+8=6x−6

8+6=6x−4x

14 = 2x

7 = x

Rectangle1:9,4

Rectangle2:6,6

Q. 5. (i) n+1

(ii) n+n+1

(iii) 2n+1=83

2n = 82

n = 41

Numbers=41,42.



Q. 6. (i) x+37

(ii) x+37+x = 141

(iii) 2x+37=141

(iv) 2x = 104

x=52

Lowergrade52, Highgrade89.

Q. 7. (i) 2y

(ii) 2y+25

(iii) y+2y+2y+25=80

(iv) 5y+25=80

5y=55

y = 11

Annieis11yearsold.

Q. 8. (i) 100 – x

(ii) 10(100 − x)+5(x)

(iii) 10(100 − x)+5(x)=650

1,000 − 10x+5x=650

1,000 −650=5x

350=5x

70 = x

Luxury seats = 100 – 70

=30

Q. 9. (i) n+2

(ii) 7n=5(n+2)+12

(iii) 7n=5n+10+12

7n=5n+22

2n = 22

n = 11

Numbers11,13

Q. 10. (i) 4.5x

(ii) (4.5x+x)−30

(iii) x+4.5x+5.5x−30=432

(iv) 11x−30=432

11x=462

x = 42

Arthur€42 Barry€189 Ciara€201

Q. 11. (i) n+1

(ii) n+n+1−5

(iii) 2n+1−5

2n −6

n −3

∴−3,−2

Q. 12. (i) 3 __ 4 y

(ii) 3 __ 4 y+y = 147

Q. 13. (i) 18,000+0.125x

(ii) 18,000+0.125x 60,000

(iii) 0.125x 42,000

x 336,000

She will have to have sales of at least €336,000

Q. 14. (i) Perhourearns12.50x

(ii) 12.50x−0.15x=12.35x

(iii) 12.50x−0.15x=2,395.90

12.35x=2,395.90

x=194

Q. 15. Correctquestion=x

Incorrectquestion=40−x

5x−3(40−x) = 112

5x−120+3x = 112

8x−120=112

8x=232

x=29

Incorrectly=40−29=11

Q. 16. x+3x+1 __ 2(3x)=550

4x+3x ___ 2=550 Amy:€100

8x+3x=1100 Brendan:€300

11x=1100 Chloe:€150

x = 100

(iii) 1.75y = 147

y = 84

Chocbars=84

Candybars=63.

⎫⎬⎭



Q. 17. 30x+30x−500+35x ____________________ 3 =2050

95x−500=6150 95x=6650 x = 70 Answer:€70

Q. 18. D = S × T

(i) Coach=55t

(ii) Car=65t

(iii) 65t+55t = 180

(iv) 120t = 180

t = 3 __ 2 or 1.5hrs.

(v) 65t–55t=75

10t=75

t=7.5

11 : 00 7:30 18 :30

Exercise 2.20Q 1. (i) x+y=25

(ii) 2x+y=35

(iii) x+y=25

−(2x+y=35)

−x=−10

x = 10

y=15 ∴Ansis10,15

Q 2. (i) x+y = 20

(ii) x – y=5

(iii) x+y =20

–(x – y=5)

2x=25

x=12.5

x+y = 20

12.5+y = 20

y=7.5

∴ Ans is 25 ___ 2 , 15 ___

 2

Q. 3. (i) 3x+2y = 11

(ii) 5x+2y = 11

(iii) 3x+6y=9

5x+2y = 11

x+2y=3

−(5x+2y = 11)

−4x=−8

x = 2

2+2y=3

2y = 1

y = 1 __ 2

SoftDrink:€2

Burger:€0.50

Q. 4. (i) x+y = 1

(ii) 4x+2y=3.40

(iii) x+y = 1

4x+2y=3.4

−(2x+2y = 2)

2x=1.4

x=0.7 ∴Pen=€0.70

y=0.3 ∴Pencil=€0.30

Q. 5. x+y = 7 Eqn 1

x = 2y+1⇒ x – 2y = 1 Eqn 2

x+y = 7

–(x – 2y =1)

3y=6

∴ y = 2

x=2(2)+1

x=5

Q. 6. (i) x+y=735

(ii) x – y=105

(iii) x−y=105

x+y=735

2x = 840

x = 420 ← Alan

420+y=735

y=735−420 y=315←Carol



Q. 7. (i) x+y = 11

(ii) 3x+y = 17

(iii) 3x+y = 17

–(x+y = 11)

2x=6

x=3

x+y = 11

3+y = 11

y = 8

∴3goals,8points

Q. 8. Let x = The number of leather sofas Let y = The number of fabric sofas x+y=50

1,000x+750y=42,500

−(750x+750y=37,500)

250x=5,000

x = 20 ← leather

y=30← fabric

Q. 9. x+y=35 ×3

3x+5y=125

3x+3y=105

−(3x+5y=125)

−2y=−20

y = 10

x+y=35

x+10=35

x=25

253mkquestion,105mkquestion

Q.10. Let x = The cost of a coat Let y = The cost of a bag x+y = 100

3 __ 5 x+1 __

 4 y=100−54

3x ___ 5+

y __ 4=46

12x+5y=920

−(5x+5y=500)

7x = 420

x=60

y = 40

∴Coat:€60

∴Bag:€40

Q.11. Let x=Thefirstsectionoftherace Let y = The second section of the race

Time = Distance _________ Speed

x __ 3+

y __ 6=30

x __ 5+

y __ 3=46

2x+y = 180

3x+5y=690

10x+5y=900

−(3x+5y=690)

7x = 210

∴ x=30

60+y = 180

∴ y = 120

First:30m

Second: 120 m

Q.12. Let x = Karl's age Let y = Eddie's age

Eqn 1 ⇒ x+y=65

Eqn 2 ⇒ 2(y−10)=x−10

2y−20=x−10

−x+2y = 10

x+y=65 x+25=65

(−x+2y = 10) x = 40

3y=75 Karl:40

y=25 Eddie:25

Q.13. Let x = Speed of one runner Let y = Speed of the other runner

(i) 2x−2y=3

(ii) 3x+3y=25.5km

(iii) 2x−2y=3 ×3⇒6x–6y=9

3x+3y=25.5 ×2⇒ –(6x+6y=51)

–12y = –42

y=3.5

2x – 2y=3

2x–2(3.5)=3

2x–7=3

2x = 10

x=5

∴Speed3.5km/hr,5km/hr.

Let x=Nosof3mquestion

y=Nosof5mquestion



Revision ExercisesQ. 1. (a) (i) −2+3=1

(ii) −2−3=−5

(iii) 3−(−2)=3+2=5

(iv) −2(3)=−6

(v) (3)2+(−2)(3)−(3) =9−6−3 =0

(b) (i) 5(−3)−1

_________ −3−1 = −15−1 _______

 −4

= −16 ____ −4 = 4

(ii) (−7)2−1

________ −7−1

= 49−1 ______ −8

= 48 ___ −8 =−6

(iii) √ ________ 

3(8)+1 _______ 

2(8) = √ 

______  24+1 ______ 16

= √ ___ 25 ____ 

√ ___ 16 = 5 __

 4

(c) (i) 1: √ _____  60 _____ 9.81=2.47seconds

2: √ ______ 

6,000 ______ 9.81=24.73seconds

3: √ ______ 

8,000 ______ 9.81=28.56seconds

4: √ ______ 

2,000 ______ 9.81=14.28seconds

Note:0.75km=750m

5: √ ______ 

1,500 ______ 9.81=12.37seconds

(ii) 2.0+2.1+1.9+1.8+2.1+2.1+1.9+2.0+2.1+1.8 ________________________________________________   10 =1.98seconds

(iii) 1.98=√ _____  2h _____ 9.81

(1.98)2 = 2h _____ 9.81

9.81(1.98)2 = 2h

9.81(1.98)2

___________ 2 = h

19.229562 = h

h≈19 m

Q. 2. (a) (i) 7x+12xy

(ii) 11a2+3a

(iii) 12x2+11x

(iv) 12y+6

(b) (i) 12x2+11xy+2

(ii) 11x2+3x−13

(iii) 2x2+21x−16

(iv) 2x2+7x−2

(c) (i) 14a3 (iv) 12x2y3

(ii) 30c3 (v) 2a2

(iii) 20a3b2 (vi) c4

(d) (i) 121a2 (iv) 27m6

(ii) 125c3 (v) −64b3

(iii) −8b3 (vi) 4k4



Q. 3. (a) (i) 6a+8b

(ii) 12a+30b

(iii) 6x+12y+18z

(iv) −4x−10y

(b) (i) 2x+6y+6x+3y = 8x+9y

(ii) 6m+18n+7m+7n=13m+25n

(iii) 3x+6y−18−2x+20y+14 = x+26y−4

(iv) 8x2+2x+6+3x2−12x−21 = 11x2−10x−15

(v) 8x2−4x−4−3x2−9x+15 =5x2−13x+11

(c) (i) 2x2−16x+x2−7x=3x2−23x

(ii) 3x2−x−3x2+x+11=11

(iii) 4a2−7ab−4a2+6ab=−ab

(iv) 6x3+2x2+2x−5x3−x2+3x = x3+x2+5x

Q. 4. (a) (i) 2 _____ x+1+

3 _____ x+7

= 2(x+7)+3(x+1)

________________  

(x+1)(x+7)

= 2x+14+3x+3 _______________  

(x+1)(x+7)

= 5x+17 ____________  

(x+1)(x+7)

(ii) 1 _____ x+5+

2 _____ x+6

= 1(x+6)+2(x+5)

________________  

(x+5)(x+6)

= x+6+2x+10 _____________  

(x+5)(x+6)

= 3x+16 ____________  

(x+5)(x+6)

(iii) 2 ______ 2x+1−

3 ______ 5x−1

= 2(5x−1)−3(2x+1)

__________________  

(2x+1)(5x−1)

= 10x−2−6x−3 _______________  

(2x+1)(5x−1)

= 4x−5 ______________  

(2x+1)(5x−1)

(iv) 12 ______ 4x+1−

3 _____ x−7

= 12(x−7)−3(4x+1)

__________________  

(4x+1)(x−7)

= 12x−84−12x−3 _________________  

(4x+1)(x−7)

= −87 _____________  

(4x+1)(x−7) or − 87 _____________

  (4x + 1)(x −7)

(v) 3 ______ 2x+1−

4 ______ 2x−1

= 3(2x−1)−4(2x+1)

__________________  

(2x+1)(2x−1)

= 6x−3−8x−4 ______________  

(2x+1)(2x−1)

= −2x−7 ______________  

(2x+1)(2x−1)

or − 2x+7 ______________  

(2x+1)(2x−1)

(b) (i) x2+6x+5

(ii) y2+9y+14

(iii) k2−6k−7

(iv) x2−11x+18

(v) 6x2+13x+5

(vi) 4a2+16a+16

(vii) 12x2−19x+5

(viii) 9y2+12y+4

(ix) 16y2−8y+1

(x) 2x(x2+6x+5)

= 2x3+12x2+10x

(c) (i) (y−6)(y+6)

(ii) x(x−50)

(iii) (x−9)(x+9)

(iv) (x+1)(x+5)

(v) (x−4)(x−3)

(vi) (x+5)(x−2)

(vii) (2x−1)(x−1)

(viii) (5x−2)(x+3)

(ix) 16(4y2−1)=16(2y−1)(2y+1)

(x) 3x(4x+5)

(xi) x(8x−11)

(xii) (2x−5)(x−4)



(xiii) 2(x2−15x+54)

= 2(x−9)(x−6)

(xiv) (5x−1)(x+7)

(xv) 5(2x2+x−1)

=5(2x−1)(x+1)

(xvi) (4x−13)(x−1)

Q. 5. (a) (i) 12y2

(ii) 36x3y

______ 6xy =6x2

(iii) 48x5y4

______ 

8x3y3=6x2y

(iv) (9p8q4)(10pq)

____________ 

15p2q3 =

90p9q5 ______ 

15p2q3

=6p7q2

(b) (i) (x+5)(x+3)

____________ x+5 = x+3

(ii) 36x3y

______ 6xy =6x2

(iii) 4x−8 _______ 2 = 4 (x – 2)

________ 2 = 2 (x−2)

or 2x−4

(iv) 6x+15 _______ 3x = 3 (2x+5)

_________ 3x = 2x+5 ______

 x

(v) 15x – 30 ________ 5x – 10 =

15(x – 2) _________ 

5(x−2) = 15 ___

 5=3

Q. 6. (a) (i) 2x=6 (iv) 3x = 12

x=3 x = 4

(ii) 7x=42 (v) 54=6x

x=6 9=x

(iii) 8x = 24 (vi) 12 = 4x

x=3 3=x

(b) (i) 5x+35=25x−25

60=20x

3=x

(ii) 8x+24=5x+30

3x=6

x = 2

(iii) 18x−27=25x−25+5

18x−27=25x−20

−7=7x

−1=x

(iv) 2x+14−5x+5=13

−3x+19=13

6=3x

2 = x

(v) 5x−1−2x−4=x

3x−5=x

2x=5

x = 5 __ 2

Q. 7. (a) (i) 5x−4=6

5x = 10

x = 2

(ii) a+3=2

a=−1

(iii) 4(x+2)+3(x+1)=4(1)

4x+8+3x+3=4

7x+11=4

7x=−7

x=−1

(iv) 2(3x+1)−5(x−1)=10(1)

6x+2−5x+5=10

x+7=10

x=3

(v) 2(5x−1)=7(x+2)+14(1)

10x−2=7x+14+14

3x=30

x = 10

(b) (i) y+t

____ p = q

(ii) p

__ v = t

(iii) ax=−by−c

a = −by−c

_______ x

(iv) t−a _____ n−1 = d

(v) A ____ 2ph = r

(c) (i) ax+kx = c

x(a+k) = c

x = c _____ a+k



(ii) r+rs=1+c

r(1+s)=1+c

r = 1+c _____ 1+s

(iii) a−b = ak

a−ak = b

a(1−k) = b

a = b _____ 1−k

(iv) a−b = c(b+1)

a−b = bc+c

a−c = bc+b

a−c = b(c+1)

a−c _____ c+1 = b

(v) a−5c = cd+d

a−d = cd+5c

a−d = c(d+5)

a−d _____ d+5 = c

Q. 8. (a) (i) 2x+y = 10

4x − y = 8

6x = 18

x =3

2(3)+y = 10

6+y = 10

y = 10

∴ x=3,y = 4

(ii) 2x+6y = 4

−(2x+y=9)

5y=−5

y=−1

x+3(−1)=2

x−3=2

x=5

      ∴ x=5,y=−1 (iii) 5x−15y = 20 −(5x−2y = 7) −13y=13 y=−1 x−3(−1)=4 x+3=4 x = 1      ∴ x = 1, y=−1

(iv) 5x−3y = 1

−(3x−3y = 0)

2x = 1

x = 1 __ 2

1 __ 2−y = 0

y = 1 __ 2

      ∴ x = 1 __ 2 , y = 1 __

 2

(b) (i) 2x−3y=2

3x+8y=3

6x−9y=6

−(6x+16y=6)

−25y = 0

                    ∴ y = 0

                    ∴ x = 1

(ii) 5x−12y = 17

2(x+2)−18(y+1)+27=0

→ 2x−18y=−13

10x−24y=34

−(10x−90y=−65)

66y=99

                     ∴ y = 3 __ 2

5x−12( 3 __ 2 ) = 17

5x−18=17

5x=35

                  ∴ x = 7

Q. 9. (a) (i) x 4, x ∈ N

      ∴ x ∈{1,2,3}

(ii) x 5,x ∈ N

      ∴ x ∈{1,2,3,4,5}

(iii) 2x 6

x 3,x ∈ N

      ∴ x ∈{1,2,3}

(b) (i) 4x 12

x 3,x ∈ Z

1 2 3 4 5 6 7



(ii) 2x 14

x 7, x ∈ N

0 1 2 3 4 5 6 7 8

(iii) 3−7 2x

−4 2x

−2 x

x −2,x ∈ R

–2–3–4 –1 0 1

(c) (i) 2x 8

x 4, x ∈ R

1 2 3 4 5 6

(ii) 15 3x

5 x

x 5, x ∈ R

3 4 5 6 7 8

(iii) 8x−24 x + 4

7x 28

x 4, x ∈ R

2 3 4 5 6 7

(iv) 4x + 5 9

4x 4

x 1, x ∈ R

0 1–1 2 3

Q. 10. (a) 44−n

(b) Adults=53−x

Female adults: 53−x ______ 2

(c) Males: ( 3 __ 5 n−200)

(d) (i) 4x + 37

(ii) 4x + 37 = 325

(iii) 4x = 288

x = 72

(e) (i) x + 5 (ii) 3(x + 5) (iii) x + (x + 5) + 3(x + 5) = 95 (iv) x + x + 5 + 3x + 15 = 95 5x + 20 = 95 5x = 75 x = 15

Marie: 15 years old.

Q. 11. (a) (i) x + y = 12

(ii) 10x + 50y = 240

(iii) x + y = 12 x + y = 12

10x + 50y = 240

_______________ 10

−(x + 5y = 24)

−4y=−12

y = 3

x + y =12

x + 3 = 12

x = 9

∴ 9 10 cent coins, 3 5 cent coins

(b) 10x + 6y = 150

5x + 18y = 150

−(30x + 18y = 450)

−25x=−300

                  ∴ x = 12

→ 5(12) + 18y = 150

18y = 90

∴ y = 5

(c) x + y = 10

2x + 3y = 23

−(2x + 2y = 20)

∴ y = 3 ← no. of 3 person rooms

           ∴ x = 7 ← no. of 2 person rooms

Exam QuestionsQ. 1. (a) 3x – 21 + 5x – 20 = 15

8x = 56

x = 7

(b) 4a + 3b = –3

5a – 2b = 25

8a + 6b = –6

15a – 6b = 75

23a = 69

a = 3

b = –5

(c) 4x – 6 + 6x 25

10x 31

x 3.1

{1, 2, 3}

ML7017_LCOL_02_AM3_Sol_v5.0.indd 45 08/04/17 11:07 AM


Q. 2. (a) 5×2=10

28 – 10 = 18

18 ÷2=9

€9perhour

(b) w=9h+2d

(c) 9(6)+12(2)+5(13.5)+2d = 161.50

145.5+2d = 161.50

2d=16

d = 8

Q. 3. (a) 3(4–5x)–2(5–6x)=12–15x–10+12x =2–3x

(b) 2–3x –6

⇒–3x –8

⇒ x 8 __ 3

x ∈{1,2} (c) −5x+4

−3x+2)15x2−22x+8

15x2−10x

−12x+8

−12x+8

g(x)=–5x+4

Or

(2–3x)(Ax+B)=15x2 – 22x +8

⇒ 2B=8and–3A=15

⇒ B = 4 and A=–5

g(x)=–5x+4

Q. 4. (a) 12 ×12 =144 cm2

(b) Length of box = 12 – 2h

Width of box = 12 – 2h

(c) V = (12 – 2h)(12 – 2h)h

V = (144 – 48h+4h2)h

V = 4h3 – 48h2+144h

Q. 5. (i) 8–6x+12=7x – 10x+35

–15=3x

x=–5

(ii) x=–5

2[4–(–15)]+12 7(–5)–5(–10–7)

38+12 –35+85

50 50

[50=50]

Q. 6. 1 __ 2 (7x−2)+5=2x+7

⇒ 7x−2+10=4x+14

⇒ 7x+8=4x+14

⇒3x=6

⇒ x = 2

Q. 7. (a) C = ( W ___ 68 ) × A

= ( 30 ___ 68 )×125

=55.147≈55

Answer:55mg

(b) (i) Formula: C = ( Y ______ Y+12 ) × A

Where C is child's dose, Y is the child's age in years, A is the adult's dose.

(ii) Formula: C = A ________ 

( Y+12 ______ Y )

C, A and Y same meaning as (i)

(iii) Formula: C = A × Y ______ Y+12

C, A and Y same meaning as (i)

(c) By algebraic manipulation each formula can be written in the same form C = YA ______

 Y+12

(d) C = YA ______ Y+12 = 6×150 _______

 6+12=50mgperday

(e) C = YA ______ Y+12

⇒  A __ 5

= YA ______ Y+12

⇒ Y+12=5Y

⇒ 4Y = 12

⇒ Y=3

OR



30 = 150Y ______ Y+12

⇒30Y+360=150Y

⇒ Y=3

The childis3yearsold.

(f) BSA=0.95(readingfromchart)

C = BSA _____ 1.73 × A

= 0.95 _____ 1.73 × 200

≈109.83mg≈110mg

(g) Young's rule: C = YA ______ Y+12

⇒  9A ______ 9+12

=90

⇒ A = 21 ___ 9×90=210mg

Clark'srule:C = W ___ 68 × A

⇒ W ___ 68×210=90

⇒ W = 90×68 _______ 210 =29

1 __ 7 kg

Weightofchildis291 __ 7 kg

C = BSA _____ 1.73 × A

⇒ BSA _____ 1.73×210=130

⇒ BSA = 130×1.73 __________ 210 ≈1.07m2

Height: 142 cm (using chart)

Q. 8. (a) 6f+2g=−3

2f+g=−2

6f+2g=−3

−4f − 2g = 4

2f = 1

⇒ f = 1 __ 2

(b) 40 −6x 19

–6x –21

6x 21

x 3.5

−5 −4 −3 −2 −1 0 1 2 3 4

3.5

5 6

(Assuming x ∈ R)

6( 1 __ 2 )+2g=−3

3+2g=–3

2g=–6

g=–3


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Chapter 2 – Algebra I