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06 CH13 Solutions of Equations (F+H) EX
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December 08, 2015
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f + 1 = 2 k + 2 = 5 q + 3 = 7
v + 6 = 13 a + 4 = 15
w + 5 = 2r + 3 = 0
g + 5 = 5
m + 2 = 2
h + 6 = 5 x 1 = 3 b 3 = 3
c 5 = 2 n 4 = 2 s 7 = 1
y 3 = 1 i 1 = 2 d 3 = 0
p 4 = 3 t + 4 = 4 z 6 = 2
e + 3 = 0 u 2 = 1 j + 12 = 12
Exercise A One step equations with addition & subtraction.The exercise starts simply and later questions are more awkward. Just they appear to be fairly simple, don't be fooled: it is easy to make errors on easy questions as well, so take your time.
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2e = 6 3j = 3 4q = 8
5t = 0 2f = 16 6w = 0
2a = 15 8y = 11 7k = 13
7v = 0 8c = 8
6d = 13
Exercise B One step equations with multiplication and division.The exercise starts simply and later questions are more awkward. Just because they appear to be fairly simple, don't be fooled: it is easy to make errors on easy questions as well, so take your time.
06 CH13 Solutions of Equations (F+H) EX
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December 08, 2015
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Exercise C Mixed one step equations.Answers should be given as simply as possible using integers, fractions and mixed numbers where appropriate.
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2e + 1 = 7 3j 1 = 8 5p + 1 = 16
4a 5 = 3 3t 4 = 17 7x + 2 = 58
6k + 5 = 11 2f 5 = 7 8u + 3 = 13
3q + 4 = 5 8b 3 = 13 4z + 3 = 9
7v 13 = 8 4m + 7 = 13 11g 13 = 9
6 3y = 6 3 + 2r = 15 32 4c = 0
5 = 3 2d 4 = 10 3h 15 = 3 9n
7 = 3w 10 21 = 1 2s 23 = 2 5i
Exercise D Two step equations.Hint: all of the equations in this exercise have integer solutions.
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December 08, 2015
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13 = 6 + 2v
24 = 4m + 7
12 = 4 3g
4f 2 = 3 5k + 4 = 18 6r 2 = 0
7a + 4 = 12
12 = 3 4s 7w + 3 = 5
3 = 6h 5 3x + 4 = 7 7 6b = 4
18y 7 = 8 5i 14 = 0 4 + 7n = 0
15 7c = 7 23 5t = 2 8z 3 = 5
8 + 7u = 1 5 9p = 4 3d + 2 = 9
7 6j = 4 13 + 7e = 3 3 2q = 0
Exercise E 2 step equations #2 Equations with noninteger solutions.Answers should be given as simply as possible using fractions or mixed numbers where appropriate.
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2e = e + 3 5k = 4k 3 2r 7 = r
2a + 1 = a + 5 5f + 3 = 4f + 9 4v 3 = 3v + 5
6m 5 = 4m + 9 3s + 1 = s 1 y 3 = 4y + 12
2w + 3 = 5w 9 2 b = 3b 6 g + 7 = 2g + 7
3 5z = 2z + 17 2 3n = 8n 9 6t 7 = 3t + 6
5h +12 = 2h +17 c 3 = 4 5c 3x 4 = 8x + 3
8 3u = 2u 3 7 2i = 6 5i 1 + 6p = p + 1
8 + d = 4 5d 9 + 4q = 4 3q 11j 4 = 1 8j
Exercise F Equations with unknowns on both sides of the equation.Hint: numbers 114 have integer solutions and 1524 have rational solutions.
06 CH13 Solutions of Equations (F+H) EX
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December 08, 2015
Exercise G Equations involving fractions.Hint: all of these examples can be solved without using brackets.
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Exercise H Equations involving brackets
06 CH13 Solutions of Equations (F+H) EX
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December 08, 2015
Exercise I Solving quadratic equations
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Solve these quadratic equations by factorising Hint: In questions 110 all signs are positive (type 1). In questions 1120, the coefficient of x is negative (type 2).In questions 2130, the constant is negative (type 3).In questions 3148, we have a mixture of types 1, 2 & 3 all muddled up.
(a + 2)(a + 1) = 0(j + 1)(j + 1) = 0 (t + 3)(t + 1) = 0
(h + 4)(h + 1) = 0 (k + 5)(k + 1) = 0 (g + 6)(g + 1) = 0
a2 + 5a + 6 = 0 i2 + 4i + 4 = 0 b2 + 6b + 8 = 0
u2 + 6u + 9 = 0 r2 6r + 9 = 0 m2 6m + 8 = 0
j2 4c + 4 = 0 c2 5c + 6 = 0 b2 7b + 6 = 0
s2 8s + 12 = 0 n2 7n + 12 = 0 v2 13v + 12 = 0
c2 8c + 15 = 0 k2 12k + 20 = 0 d2 + d 2 = 0
w2 w 2 = 0 m2 m 6 = 0t2 + t 6 = 0
p2 + 4p 12 = 0d2 d 12 = 0e2 + e 12 = 0
n2 11n 12 = 0w2 + 11w 12 = 0x2 4x 12 = 0
q2 + 2q 15 = 0 f2 2f 15 = 0 e2 11e + 24 = 0
y2 10y + 24 = 0 x2 + 2x 24 = 0 v2 5v 24 = 0
r2 10r + 25 = 0 y2 8y + 16 = 0 g2 + 9g + 20 = 0
p2 + 12p + 20 = 0 w2 w 72 = 0 z2 + z 72 = 0
h2 14h 72 = 0 z2 + 34z 72 = 0 s2 6s 72 = 0
f2 3f 54 = 0 q2 15q + 54 = 0 i2 + 29i + 54 = 0
06 CH13 Solutions of Equations (F+H) EX
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December 08, 2015
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Exercise J: Quadratic equations. (a > 1)Solve the following equations by factorisation.Hint: when doing numbers 1318, bear in mind that successive pairs of equations are connected. See if you can spot the pattern.
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Exercise K Quadratic equations using the formula.Hint: When using the quadratic formula, always write down the formula and the values of a, b & c before doing anything else.
Solve the following equations, giving your answers to 3 significant figures.
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December 08, 2015
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Exercise LSolving quadratic equations.Hint: in numbers 112, there is a pattern between each successive pair of equations. Try to spot it.
Solve these equations giving your answers in exact form.
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Exercise M Mixed quadratic equations.Hint: Many of the following examples are not in the standard ax2 + bx + c = 0.
You should rearrange them before trying to solve them.
Solve the following equations, giving your answers in exact form.
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December 08, 2015
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Exercise NSolving Simultaneous Equations #1Hints:1. Questions 1 12 can be done using addition.2. Questions 1324 can be done using subtraction.3. All questions in this exercise have integer solution sets.4. No balancing is required.
Solve the following sets of simultaneous equations using Gaussian elimination
Exercise OSolving Simultaneous Equations #2Hint: Sometimes simultaneous equations can be set out in such a way as to make substitution the simplest method.
Here are some examples of this type of problem.
Solve the following sets of simultaneous equations using substitution
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December 08, 2015
Exercise PLinear Simultaneous Equations #3 balancing one equation.
Exercise QLinear Simultaneous Equations #4 balancing both equations.
Solve the following sets of simultaneous equations.
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Solve the following sets of linear simultaneous equations
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December 08, 2015
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Exercise RSimultaneous equations involving one linear and one quadratic equation.Hint: Remember to use substitution with these simultaneous equations. Gaussian elimination will not work.
Solve the following sets of simultaneous equations.
Exercise SSimultaneous equations involving one linear equation and one curve.Hint: Remember to use substitution with these simultaneous equations. Gaussian elimination will not work.
Questions 115 can be solved by factorising, the rest require the quadratic formula or completing the square.
Solve the following sets of simultaneous equations.
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06 CH13 Solutions of Equations (F+H) EX
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December 08, 2015
Exercise E1. f = 2. k = 3. r = 4. v = 5. a= 6. g = 7. m = 8. s = 9. w = 10. h = 11. x = 12. b = 13. y = 14. i = 15. n = 16. c = 17. t = 18. z = 19. u = 20. p = 21. d = 22. j = 23. e = 24. q =
Exercise C1. e = 4 2. j = 10 3. p = 60 4. t = 3 5. x = 6. f = 987. y = 0 8. a = 30 9. u = 2 10. g = 24 11. k = 12. q = 13. v = 4 14. z = 0 15. b = 2 16.m = 17. r = 0 18. h = 19. c = 11 20. i = 21.w = 0 22. s = 11 23. n = 7 24. d = 9
Exercise B1. e = 3 2. j = 1 3. q = 2 4. t = 0 5. f = 8 6. w = 07. a = 8. y = 9. k = 10. r = 6 11. u = 50 12. g = 4813.m = 16 14. b = 4 15. z = 7 16. h = 9 17. x = 0 18. s = 3619. v = 0 20. n = 27 21. c = 1 22. i = 0 23. d = 24. p = 25
Exercise H1. e = 1 2. j = 4 3. p = 1 4. t = 5 5. f = 0 6. a = 27. k = 7 8. w = 9. q = 4 10. b = 2 11. u = 12. g = 113.m = 7 14. r = 3 15. x = 10 16. h = 3 17. c = 21 18. v = 19. y = 20. s = 21. z = 22. n = 23. i = 24. d =
Exercise G1. k =8 2. r = 3 3. v = 4 4. f = 5 5. y = 14 6. m = 27. a = 5 8. w = 9. g = 1 10. n = 2 11. z = 4 12. s = 113. h = 3 14. b = 15. x = 16. t = 12 17. i = 18. c = 5019. p = 20. d = 21. u = 3 22. j = 23. x = 24. e =
Exercise F1. e = 3 2. k = 3 3. r = 7 4. a = 4 5. f = 6 6. v = 87. m = 7 8. s = 1 9. y = 5 10. w = 4 11. b = 2 12. g = 013. z = 2 14. n = 1 15. t = 16. h = 17. c = 18. x = 19. u = 20. i = 21. p = 0 22. d = 23. q = 24. j =
ANSWERSExercise A1. f = 1 2. k = 3 3. q = 4 4. v = 7 5. a = 11 6. g = 07. m = 0 8. r = 3 9. w = 7 10. h = 11 11. x = 4 12. b = 613. c = 7 14. n = 6 15. s = 8 16. y = 2 17. i = 1 18. d = 319. p = 1 20. t = 0 21. z = 4 22. e = 3 23. u = 1 24. j = 24
Exercise D1. e = 3 2. j = 3 3. p = 3 4. a = 2 5. t = 7 6. x = 87. k = 1 8. f = 1 9. u = 2 10. q = 3 11. b = 2 12. z = 313. v = 3 14.m = 5 15. g = 2 16. y = 4 17. r = 6 18. c = 819. d = 1 20. h = 2 21. n = 2 22. w = 1 23. s = 11 24. i = 5
06 CH13 Solutions of Equations (F+H) EX
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December 08, 2015
d = 1,
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Exercise Jj = 1, r = 1,
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1.7.13.19.25.31.37.43.
2.8.14.20.26.32.38.44.
3.9.15.21.27.33.39.45.
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6.12.18.24.30.36.42.48.
j = 1a = 2, 3j = 2c = 3, 5e = 4, 3q = 5, 3r = 5h=18, 4
a = 2, 1i = 2c = 2, 3k = 2, 10d = 4, 3f = 5, 3y = 4z = 36, 2
t = 3, 1b = 2, 4b = 6, 1d = 2, 1p = 6, 2e = 8, 3g = 4, 5s = 12, 6
h = 4, 1u = 3s = 2, 6w = 2, 1x = 6, 2y = 6, 4r = 2, 10f = 9, 6
k = 5, 1r = 3n = 3, 4t = 3, 2w = 12, 1x = 6, 4w = 9, 8q = 9, 6
g = 6, 1m = 2, 4v = 1, 12m = 3, 2n = 12, 1v = 8, 3z = 9, 8i = 2, 27
Exercise I
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Exercise Kf = 0.268, 3.73u = 0.562, 3.56v = 0.732, 2.73w = 1.64, 2.14i = 2.26, 0.591j = 0.790, 0.362
a = 0.438, 4.56g = 0.372, 5.37h = 7.53, 0.531c = 0.703, 2.37z = 2.12, 0.786p = 0.690, 0.290
k = 0.697, 4.30b = 0.204, 9.80r = 0.472, 8.47n = 1.10, 0.905d = 1.22, 0.820t = 2.62, 0.382
q = 1.56, 2.56m = 0.162, 6.16y = 2.79, 1.79s = 0.208, 3.21x = 2.57, 0.907e = 1.41, 0.592
Exercise L
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Exercise M
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1.5.9.13.17.21.
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Exercise Ng = 3, h = 2i = 7, j = 2j = 5, k = 1x = 1, y = 1e = 4, f = 3g = 2, f = 1
a = 8, b = 5b = 4, c = 2c = 1, d = 2m = 4, n = 2p = 8, n = 3p = 3, q = 2
d = 10, e = 5v = 1, w = 3k = 1, m = 3d = 2, e = 3y = 3, z = 2f = 7, g = 4
p = 9, q = 8q = 2, r = 4s = 2, t = 3t = 1, n = 1u = 3, v = 3v = 5, w = 3
06 CH13 Solutions of Equations (F+H) EX
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December 08, 2015
i = , j =
i = , j =
a = , b =
a = , b =
y = , z =
y = , z =
w = , x =
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y = , z =
y = , z =
Exercise S
n = 0, p = 2n = 2, p = 0q = 3, r = 4q = 4, r = 3u = 6, v = 4u = 8, v = 2q = 6, r = 4q = 1, r = 3
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e = 0, f = 4e = 4, f = 0g = 0.5, h = 0.5g = 1, h = 0i = 6, j = 4i = 4, j = 2g = 3, h = 0g = , h =
s = 2, t = 3s = 3, t = 2n = 2, p = 2n = 4, p = 8s = 3, t = 4s = 5, t = 2u = , v =
u = , v =
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Exercise Of = 1, g = 2g = 2, h = 6x = 2, y = 5u = 1, v = 1v = 1, w = 2j = , k =
k = 1, m = 3m = 8, n = 2n = 4, p = 7z = 2, y = 8d = 1.5, e = 2.5x = 27, w = 57
a = 2, b = 4b = 3, c = 9d = 2, c = 8p = 1, p = 2q = , r = f = 0.5, e = 5.5
s = 4, t = 8t = 2, u = 4h = 2, i = 7i = 2, j = 5z = , a = 6s = , r =
1.5.9.13.17.21.
2.6.10.14.18.22.
3.7.11.15.19.23.
4.8.12.16.20.24.
Exercise Pn = 1, p = 1q = 2, r = 4s = 1, t = 3u = 4, v = 3w = 0.5, x = 3w = 1, x = 1
a = 1, b = 2g = 3, h = 2i = 5, j = 4n = 6, p = 2s = 0.5, t = 1.5y = 6, z = 8
e = 2, f = 1u = 5, v = 3e = 2, f = 1g = 2, h = 3i = 10, j = 2m = 9, k = 0
c = 4, d = 3c = 2, d = 5k = 4, m = 3q = 0.5, r = 2y = 3, z = 7a = 2, y = 7
1.5.9.13.17.21.
2.6.10.14.18.22.
3.7.11.15.19.23.
4.8.12.16.20.24.
Exercise Qh = 1, f = 1i = 3, g = 2j = 1, t = 3p = 2, v = 3s = 4, c = 2y = 3, j = 5
a = 1, k = 2b = 2, m = 3p = 4, b = 1k = 5, i = 2m = 4, q = 4r = 1, z = 3
n = 2, s = 1c = 2, h = 1d = 1, n = 4r = 2, u = 3x = 3, d = 2g = 7, e = 5
t = 2, w = 1u = 4, a = 3q = 1, n = 4e = 3, x = 2f = 8, v = 4w = 3, y = 7
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c = 3, d = 3c = 3, d = 3s = 2, t = 4s = 4, t = 4u = 1, v = 5u = 4, v = 20e = 4, f = 3e = 3, f = 10g = 2, h = 0g = 7, h = 30i = 7, j = 110i = 9, j = 30
a = 1, b = 4
c = 5, d = 12c = 3, d = 12e = 1, f = 5e = 5, f = 29e = 4, f = 6
k = 3, m = 8k = 5, m = 18p = 3, q = 11p = 6, q = 20
e = 4, f = 9e = 3, f = 9g = 0, h = 0g = 1, h = 1x = 2, y = 2x = 3, y = 7u = 1, v = 5u = 2, v = 7x = 2, w = 6x = 0.5, w = 1x = 1.5, y = 7x = , y =
Exercise R
