Solving Equations (2) DB

BOOK TWOSolving Equations

PHASE 4 LEVEL 3

ST NINIAN’S HIGH SCHOOL

MATHEMATICS DEPARTMENT

1

2

Revision

Solve the following:

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

(q) (r) (s) (t)

a − 8 = 14 45 = − 19 + z 14 = − y − 5 −16 = y − 12

5 + y = 22 14 = 4 + x −96 + k = 49 y − 2 = − 13

−5 − b = − 37 −7 = − 2 − x x − 13 = 14 0 = p − 12

51 + x = 42 −192 = k + 99 7 = 14 − x −9 + k = 15

−t − 26 = 55 29 = t + 10 −1 − p = 41 −53 = − 10 + k

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

(q) (r) (s) (t)

14 = 2m −9 = 3h12

y = 5 −5 = −x6

6q = − 12 5 =14

x14

x = − 3 −15

f = 20

−7 = 7n −48 = − 8x 2 = −m3

13

x = 6

9x = − 45 −3p = 20 50 = − 7x −x5

= − 2

−5x = − 42 6 = −2t3

25 = 10r14

= 1.5



(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

(m) (n) (o) (p)

(q) (r) (s) (t)

−3 + 7x = 39 3x + 5 = 2913

z − 2 = 25 0 = 12x + 12

8 −12

x = 9 14 = − 7 +14

x −3 =y9

+ 1 −6x + 3 = 21

−6 + 10x = 44 6 = 8 −y5

12 − 3x = 15 7 = 3 −32

n

14 = 2x − 1 −6 −3b5

= 4 −10t3

− 5 = 15 6 = − 7 + 4x

−5x = − 42 5 = −34

x + 2 −76

d + 4 = 11 20 = 3 + 6x

1

2

3

Solving Equations (1)

3

Understanding Balance Forming Algebraic Equations from Diagrams

1 Write down the equation shown in each diagram:

xd)

ae)

yc)

−cg) h)

xi)

e

j) k) l)

xa)

kb) hc)

e

xxxx kk

h h

x x xxx xx

a ay

yy

−c−c

mm

−m

xx x−e

−m−m

−m

m

−m−m

13

2 Write down the equation shown by each bar model:

a) b) c)

d) e) f)

x 7x

x x a 5a

a a23a8

d 2d

d d

b 1b

b b11b

c 9c

c c18c e 5

ee

11eee

e e

4

Understanding Balance Forming Algebraic Equations from Diagrams

3 Write down the equation shown by each angle diagram:

a) b) c)

d) e) f)

g) h) i)

g) h)

3x∘ x + 50∘

2x + 10∘

x + 40∘2x∘

x + 70∘

x +10

∘

2x + 50∘

x − 10∘

12

x∘

x + 120∘

3x + 40∘

5x − 20∘

2x + 40∘

4x∘ x + 60∘

5y 3y + 8

y + 72y + 1

z + 9

3z−

2

5

Understanding Balance Balancing Numerical Scales

The mass (in grams) is given in each of the following diagrams. Find the mass of the by ensuring ensuring balance on the scales at all times.

Δ

a)

18 ΔΔ4 Δ

Δ

b) c)Δ

Δ 8Δ

Δ3

d)

ΔΔ8 Δ

Δ

e) f)

Δ 5Δ

Δ4Δ

Δ 2 31

g)

ΔΔ3 Δ

Δ

h) i)

Δ 6Δ

Δ8Δ Δ

107

Δ

j)

ΔΔ8 Δ

Δ

k) l)

Δ 10 ΔΔ

7Δ4

12Δ

Δ Δ ΔΔ

6 Δ

m)

ΔΔ7

n) o)

Δ 5 ΔΔ

1Δ 1412 Δ

Δ ΔΔ

3 ΔΔ Δ

ΔΔ

6

Understanding Balance Relations

Using the equation shown, fill in the missing parts of the following equations to maintain balance:

1.

so (a)

(b)

(c)

4.

so (a)

(b)

(c)

9.

so (a)

(b)

(c)

2a − 3b = 4

10a − 15b =

a − 112

b =

3a − 412

b =

12g = 18h − 30

10g =

5 − 9h =

12

g =

5(a + 2b) = 80

2a + 4b =

12

a + b =

a + 2b =

2.

so (a)

(b)

(c)

5.

so (a)

(b)

(c)

10.

so (a)

(b)

(c)

4c + 6d = 7

8c + 12d =

6c + 9d =

14c + 21d =

3a − 6b = 75

2(a − 2b) =

12

a − b =

a − 2b =

112

(4n − 10p) = 9

2n − 5p =

2(3n − 712

p) =

20p − 8n =

3.

so (a)

(b)

(c)

6.

so (a)

(b)

(c)

8e − 12f = 16

6e − 9f =

e − 112

f =

6f − 4e =

5(2k − 3m) = 60

4k − 6m =

k −12

m =

9m − 6k =

7

Solving Equations with Variables on Both Sides Equations of the form ax + b = cx

Solving Equations with Variables on Both Sides Equations of the form ax + b = cx + d


(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

9t + 8 = 5t 5x = x − 16 y = 10 − 6y 8 − 3p = − 7p

3x + 13 = x 9x = 8x + 1 12 − b = 5b 7x − 9 = 5x

5x =− 7 − 3x 6x = 3x + 36 3 − 4g = 2g 4x − 27 = x


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

4x + 3 = 2x + 9 4x + 14 = − 1 + 9x 22 + 3x = 9x − 2

3x + 2 = x + 18 7 + 4r =− 13 + 8r 6x + 1 = 19 + 4x

−7 + 10x = 8x + 8 10x − 9 = 7x + 12 1 + 6x = 2x + 23

4x + 8 = 26 + x −3 + 4t =− 9 + 7t −31 + f = − 3 + 5f

Solving Equations with Variables on Both Sides Equations requiring simplification

Simplify both sides of the following equations then solve them:

(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

8x + 3 + 4x + 5 = 5x + 9 + 5x + 11 7x + 2 + 3x + 9 = 4x + 12 + 3x + 8

3x + 9x + 7 + 4 = 6x + 17 + 14 + x 9x − 5x + 8 − 3 = 7x + 13 − 5x − 2

7x + 3x − 6x + 9 = 5x + 15 − 4x 8x + 5 − 3x − x = 4x + 2x + 14 − 5x

7x + 6x − 2x − 5x = 16 + x + 9 2x + x + 7 − 3 + 5 = 12 − 4 + 7 − 3x

8x − 3x + 5x − 4 = 4 − 2x + 6 − 2 6x − 2 + 3x − x = 1 − 4x + 20 − 5

2

1


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

4r + 12 = 48 − 2r 13 − 4y = 2y − 11 5k + 4 = 44 − 3k

7b + 3 = 93 − 2b 9 − 3s = 4s + 30 14 − 6b = 4b + 34

4 − 5x = 9x + 18 7 − a = 4a − 18 50 − 3z = 2z + 15

3 − x = 9 − 3x 4 − 3x = 12 − 5x 10 − 7x = 1 − 4x

Solving Equations with Variables on Both Sides Mixed Exercise


Solving Equations with Brackets Equations of the form a(bx + c) = dSolve the following:

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

3(2x + 3) = 14 2(y + 13) = 32 2(18 − x) = 18

300 = 2(100 − x) 20 = 5(x − 4) 3(x − 1) = 18

5(5 − x) = 5 21 = 3(3x − 2) 24 = − 2(−4x + 6)

5 =− 5(−1 + 6a) −2(g − 3) =− 12 −4(3x + 5) =− 32

Solving Equations with Brackets and Collecting Like Terms


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

2(x + 3) − 1 = 7 2(x + 3) − 4 = 6 6(p − 1) + 11 = 17

2 + 5(x − 1) = 12 3(3x + 5) + 6 = 3 15 + 5(3y − 2) = 35

5(2x − 3) − 4 = 21 2 + 3(1 + 2y) = 23 11 + 2(6 − 2x) = 7

20 = 19 − 2(3x − 1) 4(5 − 2z) − 5 = 39 15 = 5 + 5(3a − 1)

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

(p) (q) (r)

(s) (t) (u)

3x = 20 − x 7m − 3 = 3m + 17 −8 + 3g = g

12 − q = 5q 5 − 3p = 4 − 2p 10x + 45 + 2x + 2 = 9x + 1 + 46

5s − 2 = 12 + 7s y = 21 − 2y −8 + 4q = 12 + 6q

−8 + 9y = 2y + 20 10 − 4y = 6 − 3y x + 22 + 2 + 5x =− 19 − x − 5

5e − 9 = 2e 7 − 12x = 15 − 8x −4x + 40 = 6x

9 − 6z = 29 − 16z 5m = − 3m + 24 8x − 16 − 3 + 3x = 1 + 9x + 18

6t + 8 = 26 − 3t 1 + 6t =− 2 + 3t 3y + 10 = y + 18

1

2 Solve the following:

(a) (b)

(c) (d)

(e) (f)

3(5x + 4) − 3x − 10 = 4x − 7 + 13 8x − 4x + 2(x − 7) = 8 + 9 − 2x + 5

9x − 6 − 6x + 7x = x + 3(x + 7) 3x + 9 + 2(x − 3) = 8x + 5x − 9x + 4

7x + 16 + 3(2x − 4) = x + 13 + 12x − 6x 6x + 6 − x − 1 = 18 + 2(2x − 3)

Solving Equations with Brackets and Collecting Like Terms

9


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

2x − 7(3 − 2x) = 11 15 − 3(x − 1) = 6 16 = 8 − 2(5 − x)

20 − 7(2y − 3) =− 1 30 = 4 − 2(3a − 1) 43 = 4 − 5(3s − 2)

28 = 3 − 5(3s − 2) 10 − (3y + 2) = 2 28 = 1 − (2g − 1)

Solving Equations with Brackets


(a) (b)

(c) (d)

(e) (f)

(g) (h)

3(x + 2) = 2(x + 5) 4(x − 3) =− 5(x + 1)

7(2z + 6) = 4(3z + 12) 2(3c − 8) =− 2(4c − 6)

3(2y + 10) = 2(5 − 2y) 9(−1 + x) = 4(2x − 3)

3(2x − 3) = 2(2x + 1) 5(3 − x) = 3(5 − x)

1


(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

(k) (l)

5(3x − 4) + 17 = 3(2x + 3) 3(3x + 5) − 7 = 5(x + 3)

2(4x − 1) + 5(x − 2) = 6(2x + 1) 8(2 + x) = 5(3 + x) − 11

3(2x − 1) + 2(3x + 1) = 7(x + 2) 3(2 − x) − 4 = 6(1 − x) + 4

6(x + 2) + 6 = 3(x − 4) 5(2x − 5) + 2 = 2(8 − 2x) + x

3(6 − d ) + 4(2d + 3) = 50 2(c − 2) − 2 = 4(2c + 1)

10(6f − 8) = 6(5f + 6) − 26 3(7 − 2t) = 3(t + 5) − 84


(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

(k) (l)

8(2x − 1) − 2(7x − 5) = 10 4(x + 5) − 3(x − 3) = 34

4(5x − 1) − 3(5x + 2) = 3x + 1 4(2x + 5) = 8 − 3(x + 7)

2 − 2(x − 3) = 4x − 4(x − 3) 3(4 − 2x) − 7(1 − 2x) = 13

8 − 2(x + 4) = 3(2 − x)12

(18x + 2) = 3(3x − 2) − 3(x − 7)

15x = 3(x − 1) − 4(1 − x) 2(x − 1) − 3(2 − x) = 4(x − 1)

7 − 5(2 − 3x) = 4(3x + 1) 5x + 6(x + 1) = x − 2(x − 3)

Multiple Brackets

10

Solving Equations with Brackets Mixed Exercise


Equations with Fractions

(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

(k) (l)

(m) (n)

(o) (p)

(q) (r)

(s) 16+3(4t-5)=7 (t)

5(z − 4) = 20 7(1 + 2x) = 0

15 = 3(4b + 3) 7(1 + 5b) =− 14

4 + 5(2x − 1) = − 41 −9 + 5(2w − 3) =− 64

−5 − (3x + 6) = − 12 −4(3g − 11) + 20 = 52

2(3y + 5) = 5(y + 1) 6(z − 3) = 3(z + 1) + 3

5(x − 3) + 6 = 10(x − 6) + 9 3(3z − 9)−4 − (5 − 2z)

3(2w − 2) + 5 = 9 − 2(3w + 1) 6(2b + 1) =− 30

4(2 − w) = 26 −20 = 5(2 + 7x)

120 = 12(3 − 2z) 2(b − 6) + 7 = 17

8(x + 3) + 7 = 4(x + 5) + 15


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

(p) (q) (r)

(s) (t) (u)

2x − 9 =x4

5x − 19 =14

x13

z = 4z + 22

−3 =9 + 3x

45 − 3x

5=− 2

8 − x3

= 3

t + 25

= 4 5 =14

(x + 7)12

(a − 9) = 3

y2

= 2y − 18x4

= 3x − 11x2

= x + 5

r6

= 3r + 85 y − 16 =y5

x2

= x − 5

2y − 1 =13

y 6m − 4 =m4

−1 =2x + 5

3

2x + 13

= 5 1 =3x − 1

52i + 1

3= 17

11

Solving Equations Misconceptions

1 Some equations have been incorrectly solved. Identify the mistake, explain the mistake so the person can fix their misconception then write out a correct version of the solution.

(a)

6(2x − 3) = 42

12x − 3 = 42

12x = 45

x =4512

= 3.75

(b)

5(2x + 1) = 55

2x + 1 = 11

2x = 12

x = 6

(c)

5x − 36

= 7

5x6

= 10

5x = 60

x = 12

(d)

4x3

− 5 = 7

4x − 5 = 21

4x = 16

x = 4

(e)

5n − 2 − (3n − 4) = 10

2n − 6 = 10

2n = 16

n = 8

(f)

3x2 − 7 = 68

3x2 = 75

x = 25

Negative variablesForming Equations

1 Write down an equation from each of the following diagrams and solve it to find the value of .x

a) b) c)


a) b) c)


a) b) c)

Shape


a) b) c)

x − 10∘

12

x∘

x + 120∘

3x + 40∘

5x − 20∘

2x + 40∘

x∘

x + 30 ∘

x + 30∘

x+

30∘

80∘ x∘

x − 20∘

x−

10∘

x∘

x + 90∘

x + 70∘

x−

10∘

2x∘x∘

2x − 10∘x + 100∘

3x + 10∘

3x + 20∘3x∘

x∘

x + 10∘ 10∘

x∘

x −15 ∘

2x∘

3x∘

x∘

12

x∘

60∘

Negative variablesForming Equations

5 Find the values of the variables given in the following isosceles triangles.

a) b) c)

6 The area of each rectangle is given in . If the lengths of the sides are in , find the value of in each rectangle.

cm2 cm x

a) b) c)

7 The perimeter of each shape is given in . If the lengths of the sides are also in , find the values of .cm cm x

a) b) c)

Section A - Shape

Forming Equations Section B - From Words

1 For each statement, write down and solve an equation to find the numbers:

(a) Trebling a number and then subtracting gives the same result as doubling the number and then adding

.

(b) Multiplying a number by and then adding gives the same result as trebling the number and then

adding .

5

2

5 3

12

3x∘ x + 50∘

5y 3y + 8

y + 72y + 1

Area = 35

x + 2

5

Area = 18

x−

33

Area = 7

4x + 2

1 2

x + 1

3

Perimeter = 15

3x − 2

3x−

2

Perimeter = 32Perimeter = 12

2x − 4

x + 2

2x−

1

Forming Equations

1 For each statement, write down and solve an equation to find the numbers:

(c) Multiplying a number by and then subtracting gives the same result as doubling the number and then

adding .

(d) Multiplying a number by and then subtracting gives the same result as adding to the number.

(e) When I add to a number I get the same result as halving the number and adding .

(f) I add to a number and then treble the answer. This gives the same result as adding to the number and

doubling my answer.

(g) I subtract from a number and then multiply my answer by . This gives the same result as adding to the

number and then trebling the answer.

(h) When is subtracted from a number and the answer doubled, the same result is reached as when the

number is halved.

6 4

20

3 2 17

4 10

5 8

2 5 7

6

2 I have pence in my pocket. John has pence more than me. Ian has twice as much as I have. Altogether we have pence.

x 2080

(a) How much in terms of have John and Ian?

(b) Write down an equation for and solve it.

(c) How many pence each have John and Ian?

x

x

3 Mr George had to pay a garage bill for the repair of his car. He spent on oil, five times this amount on labour and more on new parts than on oil. If the total bill was for , find the value of . Also find how much he spent on labour and how much on new parts.

£z£25 £88 z

4 I am thinking of a whole number .n

(a) Write down the next whole number bigger than .

(b) If these two numbers add up to , find the value of .

n

29 n

Section B - From Words

7 A boy has marbles. If he wins more, he will have three times as many as when he started, Find the value of .

x 20x

8A cricketer scores runs in his first match. He scores runs more than this in his second match and runs more than his first score in his third match. If he scored altogether in these three matches, find the value of

.

24 x 3x92

x

9 Mrs Harris bought hyacinth bulbs at each and four times as many crocus bulbs at each. If she spent altogether, how many of each type of bulb did she buy?

x 40p 5p£4.20

(a) How many girls are there?

(b) If the number of girls is twice the number of boys, find the value of .x

1 A class has pupils of whom are boys.27 x

Forming Equations Section C - From Words

(a) How many men are employed?

(b) If the number of men is four times the number of women, find the value of .y

2 A factory employs people of whom are women.150 y

(a) How many boys are members?

(b) If the number of girls is four more than the number of boys, find the value of .z

3 A youth club has members of whom are girls.30 z

(a) If there are boys, how many girls are there?

(b) If the number of boys is more than the number of girls, find the value of .

x

48 x

4 A school has pupils.960

Forming Equations Section D - Surface Area (Optimisation Introduction)

1 A rectangular channel is across and high. It is long. The total

area of the three rectangles which make the channel is . Find the value of .

20cm10cm (x + 40)cm

8000cm2 x

2 Four rectangles each have a height of and a width of . They make a rectangular tube, open at both ends as shown. The total area of the outside surface of the tube is . Find the value of .

8cm(x + 2)cm

96cm2

x

3Four isosceles triangles of base and height

are sellotaped to the edges of a square of side . The tips of the triangles are brought to a point to make a pyramid. If the total surface area of the pyramid (including base) is

, find the value of .

6cm(3x − 2)cm

6cm

120cm2 x

(x + 40)cm

10cm

20cm

(x + 2)cm

(x +2)cm

8cm

6cm6cm

6cm

(3x − 2)cm

Negative variablesEnrichment

Solutions to Equations1Put the numbers , , and into the places below:

How many different equations can you make?

How many different solutions are there?

Use four other numbers and repeat the process; do you get the same number of possible solutions?

1 2 3 4

□ x + □ = □ x + □

Box Equations2Choose four numbers to go in the boxes below:

Solve the resulting equation.

Can you choose number so your solution is an integer?

Can you choose the numbers so your solution is a particular integer?

Can. You close the numbers so you solution is , or , or ?

What if you put an in one of the boxes?

What about one of , , , in one of the boxes?

What about an in one of the boxes?

□ x + □ = □ x + □

12

27

. . .

x

2x 3x 4x . . .

x2

A Magic Square3Make all the rows, columns and

two diagonals add up to the same

number.

19 96

1

Choosing Variables4Choose three variables, say , and .

Using any operations (as many times as you like) make some equations.

e.g.

How many different equations are possible where just , and are used once and only once and no

numbers are involved?

How do you know you have them all?

Group these equations so that any one equation within the group could be obtained from re-arranging

any other equation within that group.

Choose four variables, or five, or six.

a b c

a + b = c c − b = a b = acab

= c

a b c

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Solving Equations (2) DB