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Content Preview
CANADIAN
MATHEMATICS 8
6.2 SOLVING TYPE II EQUATIONS
, , A Type II equation
will require
you to do two opposite
or inverse operations
to solve for the
(
variable (letter).
Always do the opposite of any addition
or subtraction
first, then proceeed
to do
the inverse operation
of any multiplication
or division, as shown in the examples below.
EXAMPLE
#1
EXAMPLE
#2
5x - 9 = 8
K
+ 7 = 9.3
5x = 8 + 9
5
5x = 17
L = 9.3 -7
5x = 11
5
5
5
..L = 2.3
x = 3.4
5
1
x = 11.5
A. Solve the following.
1. 3x + 2 = 14
2. L
+ 4 = 10
3. x + 3 = 3 4/5
-3
-,8
't
-5
4. 1/2 x - 8 = 16
5. 5x - 0.9 = 2
6.
1/2 x + 3/4
= 7/8
i~
-L
O.5&'
Lf
7. 1.2x - 1.2 = 8.4
8. 2/7 x - 5 = 15
9. ~ - 8 = 10
<i
,0
7
10. 4x + 8 = 4
11. 4/5 x = 60
12. 0.2x + 0.3 = 1.4
-I
15
S.5"
13. K + 3 = -8
14. 12 = K + 5
15. ]!.. + 5 = -12
5
- c:
8
_~...J
5<0
8
-13f.p
16. -18x + 5 = 13.2
-
17. 5x - 3 = 18
18. K - 8 = -10
7
- 0 ..4 5
19. 14 = 3x - 5
20. K + 18 = 5
2l.
4x + 7 = 0.07
Cot
5
-b5
-I. 13~5
22. 2/3 x - 8 = 8
23.
-13 x + 3 = 6
24.
-4.2x + 4 = -8
.;ti
-~
a ..s5"7
/3
....-r

~,
""..'t.I"I,....UIl,....I"IIn"".
nc'v~t
n ..;;, 0
~
B. Extra Practice.
Solve the following.
e
@II
l.
3x - 5 = 10
2. x + 3 = 7
3. 4x - 5 = 7
-
(
~
5
6
;;(i.(
3
~
4.
-Sx
- 6 = 13
5. 8x + 3 = -4
6. K + 6 = -9
~
-3*&
-2-
6
-70
8-
v.~)
7.
-4x - 9 = -113
8.
x + 6 = -9
9.
-3x + 7 = 21
(J
-3
;!...
~l,o
e
ti5
-iy
~
10. 4/3 x + 7 = 50
11. 0.6x + 0.3 = 0.1
12. 5/3 x + 7 = 42
_.1-
e
3~~
;(1
-3
@I
13.
-8.2x + 5 = 17
14. 27 = -9x + 9
15. 7/3 x - 6 = 6
@i
-I. '-Ho ~l+
-;.l.
5~
@
16. 2/3 x + 5 = 5
17.
Sx - 3 = 20
18. -3/ 4 X + 6 = -11
@l
0
t-J .. fo
Cl;;J... <0
~
e
19. 3.Ix + 4 = 16
20.
-3/ 9 X + 1/ 3 = 4/ 3
2l.
-o.6x + 7 = 14
-
@l
3.g101Co***
-3
- II Co
0
I}:
22. 0.8 x + 5 = 2
23.
15x + 4 = 10
24.
10x + 0 = 10
e
0.6
-;).;).5
O ..
t..t
I
t
fl
25. 3x - 5 = 6
26.
-7x - 7 = -7
27. 2x + 2 = 2
~
-
0
e
3*6
D
tf
28. 3/ 4 X + 5 = 20
29.
-3x - 8 = 20
30. 20x - 5 = 18
,
~
;zo
/./5
-73
II
31. 3x + 2 = 11
32. x - 8 = -10
33. 4x + 8 = 4
e
3
6
-/:l.
_I
~
*
34. x - 3.0 = 3 4/5
35. 4/5 x = 60
36. 0.2x + 3.1 = 4.1
~
,.
eot
-,5
S
~
,.
37.
K + 5 = -1.2
38.
-3a + 9 = 27
39. 5x - 3 = 18
8
-<0
,.
Lf .. ~
- Y 1, (p
40.
14 = 3x - 6
4l.
27 = 5 + 2x
42. 35 = -3x + 2
~
Co.y
f{
-1/
@
@
43. 3/4 X - 5 = 20
44. 0.5 = 0.5x - 10
45. 40x + 80 = -120
33t
;tl
-5
f.
~
46.
17x + 19 = 19
47.
-x + 13 = 19
48. 2x - 7 = 0.7
*
f
0
-to
3.gS
e
,
174
I
CHAPTER
6~
1
*

CANADIAN
MATHEMATICS S
6.3 SOLVING
TYPE III EQUATIONS
A Type III equation
is one where there is more than one group of variables
(letters).
To solve a
Type III equation you should first collect and place all the variables
(letters)
on the left side of the
equal sign or equation,
and all the constants
(numbers)
on the right side of the equal sign or
equation
by using opposite
operations.
That means if an x'term
is on the right side of the equal
sign, bring it over to the left side by using the opposite sign. Likewise,
if a number is being added
or subtracted
on the left side of an equation,
bring it over to the right side by using the opposite
sign as shown in the two examples below.
EXAMPLE
#1
EXAMPLE
#2
5x - 3 = 3x + 14
6x - 4.3 + 3x = 86.2 - Ix
5x-3x=14+3
6x + 3x + 1x = 86.2 + 4.3
2x = 17
lOx = 90.5
2x = 11
lOx = 90.5
2
2
10
10
x = 8.5
x = 9.05
A. Solve the following.
1. 3x + 8x = 44
2. 7x - 3x = 44
3.
-2x - 8x = -100
f I
10
~
4. 1/3 Y + 2/3 Y = 14
5.
-3.2a + 5a = 90
6. 12 = 3/4 X + 5/4 X
~\~
ILj
5D
b
7. 7x - 5x = -20
8. 3x - lOx = 15
9. 4x + 3.2x = 18
-L
-/0
~,,5
- :l. I
10. 7x + 6 = 12
. 11. 5x - 3 = 2x + 8
12.
-5x - 3 = -2x + 8
b
33
2...
-
-33"
7
13. -2/ 3 X + 5 = 7
14. 4x + 3 = 9x + 7
15. 1/2 x + 2 = 3/4 X + 7
'i
-3
-.;tD
--5
16. O.4x + 8 = 0.6x - 7
17. 6x + 1.2 = 9x + 2
18. 6/7 X - 2 = 2/7 x + 8
-
,5
-D ..~<O
/7.'5
19. 1/ 3 X + 20 = x + 4
20.
1/ 2 a + 9 = 2a - 6
21. O.02x + 7 = 0.03x + 5
;t'i
/0
~DO
22.
Sx - 10 = 25x + 3
23. 7x - 10 = 3x + 14
24. O.ly + 0.01 = 0.01y + 0.1
-u
Co
I
~o
25. 3x + 10 + 5x = 90
26. 6x - 5 - 2x = 7 + 1
27.
16 + 45x = 39x + 310
I
{O
I:.
3*~5
Lf~
28. 7x - 3 - 2x = 8
29. 8x + 4x + Sx = 7
30. 2x - 3 = 6x + 4x
I
cl.~
-2-
-
17
~
....-r


CANAD!AN
MATHEMATU:S 8
6.4 SOLVING
TYPE IV EQUATIONS
ir.
A Type IV equation
has one or more sets of brackets or parentheses.
We use the number on the
outside of the brackets
as a multiplier
and multiply it by everything inside the brackets.
You then
proceed
to solve the equation
the way you would have done if it was a Type HI or a Type n
equation.
The examples
below show the steps in the procedure.
EXAMPLE
#1
EXAMPLE
#2
3(x + 7) = -39
4(2x - 5) = 2(3x + 6)
3x + 21 = -39
8x - 20 = 6x + J2
3x = -39 - 21
8x - 6x = 12 + 20
3x = -60
2x = 32
3x = -60
2x = 32
-
--
3
3
2
2
x
= -20
x = 16
A. Solve the following.
1. 3(x - 5) = 6
2. 4(2x + 3) = 5
3. 7(3x - 5) = 2(x - 3)
4. 4(3x - 5) + 2(x + 3) = 1
5. 6(3x - 5) = 12
6. 4(2x - 8) = 16
7. 7(x + 3) = 10
8. 4(x + 2) = 5(x - 3)
9. 7(x - 2) = 5(3x - 8)
10. 4(2x + 6) - (3x + 4) = 7
11. 6(2x - 8) = 12
12. 7x - (2x + 4) = 5
13. 6(3x - 5) = 7
14. 3(x + 2) = 20
15. 4(2x - 1) = 4
16. 4(x + 2) = 3(x - 3)
17. 7(2x - 3) = 5(2x + 8)
18. 3(4x - 5) = 2(3x - 2)
19. 5( -3x + 2) = -7
20. 4(2x + 3) + 2(x + 5) = 7
21. 6(2x - 4) + 2(x + 3) = 7
22. (x + 2) + (x - 3) = 5
23. 0.4(3x + 8) = a.2(x + 5)
24. 0.5(8x - 4) = 10
25.
1/2 (4x + 8) = 3/4 (I2x
+ 16)
26. 3(5x - 3) + 4(2x - 8) = 0
27. 5/7 (l4x
- 35) = 10
28. 12(0.5x - 0.3) = 18
29. 3/5 (lOx - 20) = 50
30. 5.6(0.5x - 0.7) = 15.68
77 I
.,.,.,

CANADIAN MATHEMATICS 8
B. Extra Practice.
Solve the following.
1. 6(3x + 2) = 15
2. 4(2x + 6) = 5(3x - 14)
3.
2(x
+ 3) = 3(7x - 5)
4. 3(a + 5) = 2(a - 5)
5. (x + 7) = (3x - 14)
6.
2(x - 3) = 21 - 6x
7.
-3(2x
+ 6) = 6(2x - 12)
8.
-4x + 6 = -3(x + 5)
9. 3/4 (l6x + 24) = 1/5 (lax + 25)
lO. 3( -4x + 8) = -10
11. 7(3x + 5) = 2(4x
- 3)
12. Sex + 3) = 6(x + 3)
13. 8(2x + 3) = 9(4x + 6)
14. 2(8x + 60) = 5(9x - 10)
15. 4/5 (lOx - 50) = 5/8 (16x - 24)
16. 2.1(x + 5) = 3.4(5x + 10)
18. 0.5(2x + 8) = O.4(lOx - 9)
20. 4(x + 3) - (2x + 1) = 7(x - 3)
21. 5(2x + 3) = 12 .
22. 0.6(lOa - 4) = 0.7(15a
+ 3)
23.
-5(2x
- 8)( + 3(x + 5) = 7
24. 4(x + 3) = 6
25. 5(x + 3) = 15 + 2x
26. 3(x) + 10 = -20
27. 4(y + 3) = 3(y + 12)
28.
-2(x
- 3) = 18
29. 4(x) + 4(3) = 4(9)
30. 8(3 + x) = 64
[ 78 I
I CHAPTER 6

CANADIAN
MATHEMATICS S
6.4 SOLVING TYPE IV EQUATIONS
"'. A Type IV equation
has one or more sets of brackets
or parentheses.
We use the number on the
f outside of the brackets as a multiplier and muitiply it by everything inside the brackets. You then
proceed
to solve the equation
the way you would have done if it was a Type III or a Type II
equation.
The examples
below show the steps in the procedure.
EXAMPLE
#1
EXAMPLE
#2
3(x + 7) = -39
4(2x - 5) = 2(3x + 6)
3x + 21 = -39
8x - 20 = 6x + 12
3x = -39 - 21
8x - 6x = 12 + 20
3x = -60
2x = 32
3x = -60
2x = 32
3
3
2
2
x = -20
x = 16
A. Solve the following.
1
1. 3(x - 5) = 6
I
2. 4(2x + 3) = 5
- ""f
,0
3. 7(3x - 5) = 2(x - 3)
I ~
4. 4(3x - 5) + 2(x + 3) = 1
I ,'Lf
5. 6(3x - 5) = 12
.:2.. 3,
6. 4(2x - 8) = 16
4
7. 7(x + 3) = 10
-
( -=j
8. 4(x + 2) = 5(x - 3)
9. 7(x - 2) = 5(3x - 8)
3. ~5
10. 4(2x + 6) - (3x + 4) = 7 - ~ .le
11. 6(2x - 8) = 12
5
12. 7x - (2x + 4) = 5
I. 8
I
13. 6(3x - 5) = 7
;{~
14. 3(x+2)
=20
t-J .. Co
15. 4(2x - 1) = 4
16. 4(x + 2) = 3(x - 3)
-11
17. 7(2x - 3) = 5(2x + 8)
/5.;;''5
18. 3(4x - 5) = 2(3x - 2)
I. ~ ~
-:;:1..
19. 5(-3x
+ 2) = -7
/ T5"
20. 4(2x + 3) + 2(x + 5) = 7
-/.5
II
21. 6(2x - 4) + 2(x + 3) = 7
I Ii
22. (x + 2) + (x - 3) = 5
3
23. 0.4(3x + 8) = 0.2(x + 5) -:l. 'l.
24. 0.5(8x - 4) = 10
3
Ig
.L
/...;--
25. 1/2 (4x + 8) = 3/4 (12x + 16) -/7
26. 3(5x - 3) + 4(2x - 8) = 0
~3
27. 5/7 (l4x
- 35) = 10
3 .5
28.
12(0.5x - 0.3) = 18
3. (p
29. 3/5 (lOx - 20) = 50
/0. ~
30. 5.6(0.5x - 0.7) = 15.68
I

t;;ANADIAN
MATHiEMATSCS
S
B. Extra Practice.
Solve the following.
1. 6(3x + 2) = 15
2. 4(2x + 6) = 5(3x - 14)
.-L
(0
13 ~
3. 2(x + 3) = 3(7x - 5)
4. 3(a + 5) = 2(a - 5)
~
1/7
-;)5
5. (x + 7) = (3x - 14)
6. 2(x - 3) = 21 - 6x
/01
.3-
3 8
7.
-3(2x + 6) = 6(2x - 12)
8. -4x + 6 = -3(x + 5)
3
~I
9. 3/4 (l6x + 24) = 1/5 (lOx + 25)
10. 3( ---4x+ 8) = -10
-1.3
;(~
11. 7(3x + 5) = 2(4x - 3)
12. 5 (x + 3) = 6(x + 3)
::L
-373
-3
l3. 8(2x + 3) = 9(4x + 6)
14. 2(8x + 60) = 5(9x - 10)
-1.5
5~
.;<C;
15. 4/5 (lax
- 50) = 5/8 (l6x
- 24)
16. 2.1(x + 5) = 3.4(5x + 10)
-1:2.5
-1.57
17. 3/4 (x + 2) = 7/8
-~
18. 0.5(2x + 8) = 0.4(10x - 9)
(p
~.53
19. 1/2 (x - 10) = 3/4 (x + 20)
20. 4(x + 3) - (2x + 1) = 7(x - 3)
-~O
Co.LJ
21. 5(2x + 3) = 12
22. O.6(10a - 4) = O.7(l5a + 3)
- 0 ..3
-/
23.
-5(2x - 8) + 3(x + 5) = 7
24. 4(x + 3) = 6
10
Co-=;
-1..5
25. Sex + 3) = 15 + 2x
26. 3(x) + 10 = -20
0
-/D
27. 4(y + 3) = 3(y + 12)
28.
-2(x - 3) = 18
;;(Lj
-lo
29. 4(x) + 4(3) = 4(9)
30. 8(3 + x) = 64
(.p
5
0r-------------
~J CHADTr::'D t::.

CANADIAN
MATHEMATICS 8
6.5 SOLVING
RATIO TYPE EQUATIONS
.r'tL
A Ratio Type equation
is classsified as one in which there are two 'fractions: one on either side of
the equal
sign.
To solve these equations
we first have to use the procedure
known as cross-
multiplication.
Here
we multiply either
the top left of the equation
by the bottom right of the
equation or we multiply ti'1etop right of the equation
by the bottom
left of the equation.
We place
one result on the right side of the equation
and the other result on the left side of the equation.
We then proceeed
as we would in a Type II, III or IV question
as shown in the examples below.
EXAMPLE
#1
EXAMPLE
#2
..L = -3
3x + 5 = l
6
2
4
2
2(x) = 6( -3)
2(3x + 5) = 4(7)
2x =
-18
6x + 10 = 28
2x = -18
6x = 28 - 10
2
2
6x = 18
x = -9
6x = l[
6
6
x = 3
A. Solve the following.
1. x = 7
2. 3x = 6
3. -9 = 6x
-
-
-
-
3
1
5
10
6
-1
4.
-6 = -12
5. K = l
6. K = -7
4
x
7
3
6
8
7. x = 5
8. 3x = 9
9. y + 4 = U 10
-
-
3
8
4
2
3
10. 2x - 3 = 3
11.
x = 4
12. 5x + 2 = 1
-
-
5
0.2
1
7x + 1
4
13.
2
=
3
14. 5x + 3 = 1
15. x + 2 = 5
1 - x
1 - 2x
7x - 2
4
3
6
16.
4x = ~
18. L=..2 = v + 3
3.5
0\2
3
6
19. 2(x + 5) = 3(x - 2)
20. 1. = 1
21. 5(3x - 2) = 2(3x + 5)
2
2
x
8
6
3
22. 2x - 6 = 3
23. 2x + 6 = 3
24. 2(x - 5) = Q.
7
4
x + 6
4
3(x + 4)
7
25. 3(x + 5) = 1
26. 3 + x = 5
n
--
27.
X (8x - 12)
(lOx + 6)
-
7(x - 2)
5
15
2
3
7
~

CANADIAN MATHEMATICS 8
L
J
,...
I
~- I
B. Extra Practice.
Solve the following.
t~,
~---I
,".
l. 6 = 5
-
-
2.
3x + 6 = ~
--I
x
8
2
4
~:
""~
i'.,~
~'I
3. -3 = -4
4. 4x - 3 = 2x + 1
,..;"""
-
-
x
8
2
6
I
~I
s. x = 5x + 2
6,
-4 = -8
I

7
1
fo-
x
-4
~I
7. 2x + 8 = 2x + 8
8. 8e4x - 3) = 2(8x + 1)
'..-
I
r-
5
9
2
3
'..I
.'
: I
9.
-sex + 3) = -2(x + 5)
10.
-x = -5
f'"
2
3
6
4
I
'<'
11. 3(x + 5) = 7(2x + 3)
C'.
12.
x = 9
C I
2
4
13
:-I
13. 7x = 6x
-
-
14. 4x - 3 = 5x - 3
C I
3
-4
5
5
C I
r'-
\.c-
I
15. 8x - 2 = 4(x + 2)
16. 4x = I
F'
3
1
\,.:~
6
8
~~', C ".,t
17. 6x + 5 = 7x + 3
18. 4ea - 6) = 9(2a + 3)
C I
2
8
3
3
C.
f"
19. 3x = -8
20 . .Q = 1
...
I
5
2
-x
5
C.
2l.
0.6x = 4x
22.
3(x +2) = 21
C.
0.3
6
7
7
C.
C
23. -.l = ---=2
24.
-.l = 2.
C II
7x
3(x + 5)
-x
5
c: II
25. 3( -2y + 5) = 1
26, 4x = 3x
C.
7
2
7
5
c ..
27.
Y4x + 2) = 1
28.
3(x - 5) = Si
C.
3
2
5
C.
C ..
29. 2x - 3 = x + 2
30. 6x - 5 = 3(x + 2)
C ..
6
4
6
4
C.
C
'\.
31. x + 2 = -x - 2
32. 5x = 3
-
-
~
5
5
t
'
4x
2
-.C_
elli
~
r
CHAPTER 6
--
~",

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