Equations
Equations
Curriculum Ready
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ISBN
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This booklet shows you how to apply algebraic skills in the solution of simple equations and problems. These words appear a lot in this unit. Investigate and write a brief description of their meaning here now. Equation
Opposite
An algebraic expression that has an equal sign ( = ) in it. Balanced
Formula
Variable
Substitute
is Give th
Q
a go!
A number plus one is multiplied by three. This is equal to seven times a number minus five. What is the value of the number?
Work through the book for a great way to solve this
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How does it work?
Equations
Keeping things in balance Like a balanced set of scales, an equation is an algebraic expression where the left side equals the right side. Left-hand side 0
1
2
3
=
4
5
6
7
8
9
LHS = RHS in equations
Right-hand side
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Write the equation represented by each of these balanced scales. (i) Left-hand side 3#a 3a
aaa 0
1
2
3
Right-hand side
= 4
5
6
7
8
9
= +1
3a = 4 (ii)
Left-hand side 6#n 6n
nnnnnn 0
1
2
3
4
5
6
7
= 8
4# 4 #+ 1 4
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
9
left-hand side = right-hand side when balanced
Right-hand side
nn
= +1
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
2 # n and 2 # 2 # n and 2 # 1 2n + 2
left-hand side = right-hand side when balanced
6n = 2n + 2 (iii) Left-hand side 2 # m and 3 # 2 # m and 3 # 1 2m + 3
m m 0
= 1
2
3
4
5
6
7
8
9
Right-hand side
m m m
= +1
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
left-hand side = right-hand side when balanced
2m + 3 = 3m + 1 (iv)
Write the new equation for (iii) if three ms are added to both sides. Right-hand side (3 + 3) # m and 1 # 6m + 1
Left-hand side (2 + 3) # m and 3 # 5m + 3 5m + 3 = 6m + 1
2
3 # m and 1 # 3 # m and 1 # 1 3m + 1
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How does it work?
Your Turn
Equations
n 0
= 1
2
3
4
5
6
7
8
9
= 1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Left-hand side
Right-hand side
Right-hand side
Equation:
1
2
3
4
5
6
= +1
d
xxx 0
= 7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Left-hand side
m m 0
1
2
3
4
5
m
= 6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Left-hand side
Right-hand side
Equation: 2
KEEPING T 0
Equation: = +1
w
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Left-hand side
c
= +1
b
Right-hand side
Equation:
Using the same balanced scales from question 1, write the new equation if these changes were made: a
= + 1 One circle added to both sides
n 0
b
2
3
4
5
6
7
8
9
w
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Left-hand side
0
= 1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Left-hand side
Right-hand side
Right-hand side
New equation:
New equation: c
= + 1 Three circles added to both sides
= 1
= + 1 One x added to both sides
xxx 0
1
Left-hand side
2
3
4
5
6
d
= + 1 Multiply both sides by 2
m m
= 7
8
9
0
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Left-hand side
Right-hand side
New equation:
1
2
3
4
5
m
= 6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Right-hand side
New equation: Equations Mathletics Passport
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../20..
. ..../..
T
= +1
a
* CE
These scales are all in balance. Write the equation each one represents.
IN BAL AN NGS HI
1
KEEPING
Keeping things in balance
* CE
IN BAL AN NGS HI
3
How does it work?
Equations
Opposite operations These are pairs of mathematical operations that do the opposite to each other. The opposite to adding is subtracting . So and are opposite operations. The opposite to multiplying is dividing . Inverse = Opposite
So and are opposite operations. Opposite operations are also called inverse operations. It means the same thing!
Use opposite operations to move clockwise (forwards) or anti-clockwise (backwards) in these puzzles. Complete these circle puzzles: (i) With numerical values only
'3 15
+ 13
#3
5
- 13
-3
2 -2
+3 8
#
+2 4
#4
1 2
16
'1 2
Pair of opposite operations
'4 (ii) With numerical and algebraic values
+4
5x - 4 3 -4
5x 3
-4
'3
+4
5x 3 '3
5x #3
#5
5x
'5
x
#5
4
I
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#3
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'5 Start here
How does it work?
Your Turn
Equations
Opposite operations 1
Fill in all the missing gaps with values and/or operations for these circular calculations. '8
a
48
#8
6
+ 23
+9
-9
15 b
+5 30
27
- 26
#2
# #
'2
7.5
2 3
18
c
- 4.5
14
+
'8 squared
144
3
#
and undo each other, and are another example of opposite operations
2
' - 14
- 12 # ^- 3h
6
' ^- 3h
#2
d
13.5
-
'2 3
#5
cubed
+
+ 7 10
- 7 10
#
' + 22 - 22
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Equations
Opposite operations 2
Fill in all the missing gaps with values and/or operations for these circular calculations. -6
a
+6
3m + 2
3m - 4 #
' m+2
6m - 8 -2
+2
+ 5m m
Start here
- 5m
-8
b
# 11
+8
' 11 +8
-8
11(2x + 8)
# 11
2x
c
#3
2+
y 3
6-y
'2 - 14
-
+ 14
y 3
(- 1 ) 3
(- 1 ) 3
' #
+8
y+8
-8 6
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2x + 8
#2
'3
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'11
2x
How does it work?
Your Turn
Equations
Opposite operations Complete these flow charts to get the variable in each expression all by itself.
3h + 6 +6
b
#3
E
5r - 1 -1
#5
RATION OPE S
0...
/2 ../ ....
...
RATION OPE S
c
11(c + 8) +8
# 11
d
1+ d 13 ' 13
e
+1
- 4m - 9 5 -9
f
'5
# (-4)
11k + 5 4 +5
'4
g
OPPOS IT
OPPOS IT
a
E
3
# 11
- 2 (5j + 12) # (-2)
+ 12 Equations Mathletics Passport
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Equations
Solving simple equations When asked to solve an equation, you are really being asked: “What value does the variable need to be to keep the equation in balance?” Simple equations like these can be solved mentally. Solve these equations: (i) a + 4 = 9
(ii)
a+ 4 = 9 `a = 5 p = 2 8 p '8 = 2
p = 2 8
` p = 16
Think “what number plus 4 will give 9”? This number plus 4 will equal 9 Think “what number does 8 go into twice”?
This number divided by 8 will equal 2
Always line up the equal signs vertically when setting out solutions Opposite operations can be used to get the variable by itself. Remember to keep the equation balanced. Solve these equations by getting the variable all by itself: (i) b - 11 = 6
b - 11 + 11 = 6 + 11 b - 11 + 11 = 6 + 11 ` b = 17 b is now by itself
(ii) 2x = 14 One Step equations
x is now by itself
2x = 14 2 2 2x 14 7 = 2 2 ` x = 7
Add 11 to both sides to keep it balanced
Divide both sides by 2 to keep it balanced
Only one opposite operation is required to solve these Solutions check Always ask yourself: Does my solution ‘feel’ correct? The best way to find out is by checking your answer to see. Is y = - 9 the solution to the equation y - 18 = 3y ? When y = - 9, y - 18 = 3y becomes: - 9 - 18 = 3 # (- 9) Replacing the variable with a number is called substitution
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Substitute in the number Check if the equation is balanced
- 27 = - 27
` the solution of y = - 9 is correct
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Your Turn
Equations
Solving simple equations Solve these equations mentally.
s + 5 = 15
`s =
b
t - 5 = 24
c
`t =
g 42 ' i = 7 18 # o = 18 `o = `i = f
h
a '6 = 6 `a =
16 + n = 52 `n =
2
5
14
25
34
45 47
Join the dots with the answers in the same order as the questions to see what shape the variables form.
19
24
e
44 ' t = 4
3 42
17
43
28
15
30
2.5 # s = 25 `s =
16 32
38
EQUATI ON LE P M
20
S
x + 7 = 16
b
` x = d
a 12 = 5
q ' 4 = 24 ` q =
c
` y = e
` a = g
y + 15 = 13
3k = 27
f
z ' (- 2) = 8
i
` z =
14w = 42
b - 13 = - 7 ` b =
15 + g = - 4 k - 2r = - 13 ` r =
l
48 ' p = 3 ` p =
Equations Mathletics Passport
....
` w =
j
` g =
...
./20 /....
` m =
` k = h
m- 3 = 8
SOLVING SI
11
Use opposite operations to solve these one step equations. a
6
27
43
`p =
S
37
p + 6 = 35
31
41
45
18 34
35
56 33
23
41
36
12
j
`t =
7
39
4
52
52
i
8
29
11 - r = 2
10
26
40
9 2
22
`r =
13
31 1
d
SOLVING SI
a
EQUATIO LE N MP
1
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How does it work?
Your Turn
Equations
Solutions check 3
Check to see if these solutions are correct or incorrect. Show your working. a
b
c
d
e
Is a = 2 the solution to the equation ? 72 ' a = 36 72 ' 2 = 36 36 = 36
CORRECT
INCORRECT
Is k = 12 the solution to the equation k - 7 = 5 ?
CORRECT
INCORRECT
Is b = 6 the solution to the equation - 4 # b = 24 ?
CORRECT
INCORRECT
Is x = - 4.5 the solution to the equation x - 8 = 12.5 ?
CORRECT
INCORRECT
Is m = 16 the solution to the equation 52 ' m = 3 1 ? 4
CORRECT
INCORRECT
CORRECT
INCORRECT
Is p = - 3 the solution to the equation 2p - 4 = - 10 ?
CORRECT
INCORRECT
Is n = 1.5 the solution to the equation 5n + 14 = 20.5 ?
CORRECT
INCORRECT
CORRECT
INCORRECT
CORRECT
INCORRECT
f Is q = 8 the solution to the equation 9 + 3q = 33 ?
g
h
i Is d = 2 the solution to the equation 8d - 6 = 5d ?
j
10
Is w = - 2 1 the solution to the equation 18 - 3w = 4w ? 2
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Equations
Two-step equations These need two different opposite operations to solve them.
Remember that what is done to one side must be done to the other.
For simple two-step equations, usually the coefficient ! 1 . Solve these equations by isolating the variable using inverse operations. (i) 2x + 3 = 7
2x + 3 - 3 = 7 - 3
Subtract 3 from both sides
2x = 4
x is still not by itself
2x ' 2 = 4 ' 2
Divide both sides by 2
` x = 2 Coefficient of x = 2
(ii) 5 + 2a = 20
5 - 5 + 2a = 20 - 5
Subtract 5 from both sides
2a = 15
a is still not by itself
2a ' 2 = 15 ' 2 ` a = 71 2
(iii) 10 - d = 22
Divide both sides by 2
10 - 10 - d = 22 - 10
Subtract 10 from both sides d is still not by itself
- d = 12 - d ' (- 1) = 12 ' (- 1)
Divide both sides by (-1)
` d = - 12
Positive divided by a negative gives a negative answer
Coefficient of d is -1 Take care with negative signs
(iv) - 4k + 14 = 3
- 4k + 14 - 14 = 3 - 14
Subtract 14 from both sides
- 4k = - 11
k is still not by itself
- 4k ' (- 4) = - 11 ' (- 4) ` k = 2 3 or 2.75 4
Divide both sides by (-4)
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Equations
3x '
=
6g '
'
= g =
4a + 11 = 39
d
5w - 12 = 18
` a = 2b + 12 = 8
e
` w= f
3n - 5 = - 8
` b = g
` n=
10 - 2m = 18 10 - 2m - 10 = 18 - 10
25 - 6p = - 29
h
- 2m = - 2m '
=
' ` p=
m = i
3- g = 8
j
15 - y = 16
` g = k
` y=
9- q = 4
l
25 = 13 - n
` q = 12
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` n = Equations Mathletics Passport
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'
0... ../2 / . ..
....
SETP TWO-
6g =
x = c
S
6g - 1 = 27 6g - 1 + 1 = 27 + 1
3x =
TION
b
3x + 5 = 14 3x + 5 - 5 = 14 - 5
a
EQUA
Use inverse operations to solve these two-step equations.
STEP TWO-
Two-step equations 1
S
TION
EQUA
Where does it work?
Equations
Equations with variables on both sides The aim is to get all the variables on one side and the numbers on the other side of the equation. Solve these equations using inverse operations. (i) 5x = x + 8
5x - x = x - x + 8 4x = 8
Subtract x from both sides Numbers and variable on different sides
4x ' 4 = 8 ' 4
Get the variable x by itself
` x = 2
(ii) 6b = 9b - 12
6b - 9b = 9b - 9b - 12 - 3b = - 12 - 3b ' (- 3) = - 12 ' (- 3)
Subtract 9b from both sides Numbers and variable on different sides Get the variable b by itself
` b = 4
(iii) 3h - 35 = 4 - 5h
3h + 5h - 35 = 4 - 5h + 5h
Add 5h to both sides
8h - 35 = 4 8h - 35 + 35 = 4 + 35 8h = 39
Add 35 to both sides Numbers and variable on different sides
8h ' 8 = 39 ' 8
Get the variable h by itself
` h = 4 7 or 4.875 8
(iv) 2p + 18 - 8p = 24 - 3p
18 - 6p = 24 - 3p 18 - 6p + 6p = 24 - 3p + 6p
Simplify by collecting like terms Add 6p to both sides
18 = 24 + 3p 18 - 24 = 24 - 24 + 3p
Subtract 24 from both sides Get the variable p by itself
- 6 = 3p - 6 ' 3 = 3p ' 3 - 2 = p or p = - 2
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Equations
SI
NS O TI UA
....
EQ
` b = c
` x =
3d = 18 - 5d
d
w + 20 = 7w - 4
` d = e
` w = f
g + 13 = 2g - 5
5m - 7 = 2m - 1
` m =
` g = g
2t - 12 = 12 + 6t
h
16 - 2k = 9 - 5k
` t =
14
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` k =
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/20 . . . . /.
S
b 18 - 4x = 2x
DE
a 10b = 5b + 20
TH
Use inverse operations to solve these equations containing variables on both sides.
RI
VA
BO
1
ES
L AB
ON
Equations with variables on both sides
TH
Your Turn
WI
Where does it work?
Where does it work?
Your Turn
Equations
Equations with variables on both sides 2
Use inverse operations to solve these equations containing variables on both sides. a 17 - 2k = k + 5
b
8 + 2g = 8 - 2g
` k = c
` g =
- x - 11 = x - 19
d
- 16 + 9w = 4 - w
` x = e
` w =
2y - 21 + 6y = 18 + 5y
f
5a + 4a + 9 = 21a - 39
` y = g
` a =
10 + 2r - 3r + 4 = r - 8
h
5p - 13 = 3p - 21 + 4p
` r =
` p =
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Equations
Equations with fractions For equations with a fraction, multiply both sides by the value in the denominator. Use inverse operations to solve these equations containing fractions. (i) 5a = - 15 4
5a 4
#4
= - 15 # 4
5a = - 60 numerator denominator
Multiply both sides by the denominator (4) a is still not by itself
5a ' 5 = - 60 ' 5
Divide both sides by 5
` a = - 12
5a = 5 # a , so another way to solve this 4 4 simple equation is to divide both sides by 5 4
(ii) m - 6 = 4 2
m- 6 #2 = 4#2 2 m- 6= 8 m- 6+ 6 = 8+ 6
Multiply both sides by the denominator (2) m is still not by itself Add 6 to both sides
` m = 14 (iii) - 5 +
y = 3 6
- 5+ 5+
y 6
y = 3+ 5 6 y = 8 6
#6
= 3#6
Add 5 to both sides y is still not by itself Multiply both sides by the denominator (6)
` y = 18 When only the variable term is in fraction form, we remove the other terms first.
(iv) n + 9 = - 2 1 4 2
16
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n + 9- 9= 4 n = 4 n #4 = 4 `n =
-21 - 9 2 - 11 1 2 - 11 1 # 4 2 - 46
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Subtract 9 from both sides n is still not by itself Multiply both sides by the denominator (4) Positive
#
negative = negative
Where does it work?
Your Turn
Equations
Equations with fractions Solve each of the equations below containing fractions. 1
a
3v = 6 4
3v 4
#4
b
7t = 21 3
= 6#4
3v =
3v '
=
'
`v = c
`t =
3u = - 12 2
d
- 6h = 4 5
`h =
`u = 2
a
x+ 4 = 6 3 x+ 4 3
#3
b
y - 12 = 2 4
= 6#3
x+ 4 =
x+ 4
=
-
` x = c
`y =
d - 16 = - 9 1 2 2
d
-5 = 8+ a 2
`d =
`a =
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Equations FR
WI
TH
Your Turn
TI UA EQ
#
` y = c
4+ h = 6 5
`k = d
m +5 = 1 7
`h = e
`m =
-3 + x = - 6 5
f
t + 11 = 5 6 2
`x =
18
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`t =
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A
k - 10 = 2 5
FR
y + 2- 2 = 5- 2 3 y = 3 y # = 3
S
b
ON
y +2 = 5 3
CT I
a
U
3
S
EQ
`p =
`b =
TH
S
f
0...
..../...../2
ON
ON
- 0.5 + p = - 6 3
WI
4 b e 12 = +3 -
TI
AT I
Solve each of the equations below containing fractions. 2
AC
ON
S
Equations with fractions
Where does it work?
Equations
Equations with parentheses This first method works best for simple equations with only one pair of parentheses. Solve these equations containing parentheses. (i) 2 (x + 2) = 6
(ii) - (2h - 5) = 4
means
2 (x + 2) ' 2 x+2 x+ 2- 2 ` x
= = = =
6 '2 3 3- 2 1
- (2h - 5) ' (- 1) = 4 ' (- 1) 2h - 5 = - 4 2h - 5 + 5 = - 4 + 5
Divide both sides by 2 Get the variable x all by itself
Remove the # (- 1) Get the variable h all by itself
2h = 1 #
(-1)
2h ' 2 = 1 ' 2 ` h = 1 2
Remember: Distributive law: a^ b + ch = ab + ac a^ b - ch = ab - ac
You can use the distributive law to expand the parentheses first.
Solve these equations after first expanding using the distributive law. (i) 3 (a - 7) = 4a
(ii) 7 = - 6 (d + 4)
#
3^ a - 7h 3a - 21 3a - 3a - 21 - 21 `a
= = = = =
4a 4a 4a - 3a a - 21
Expand the LHS Subtract 3a from both sides
#
7 7 7 + 24 31 31 ' (- 6) ` -51 6
= = = = =
-
6^ d + 4h 6d - 24 6d - 24 + 24 6d 6d ' (- 6)
Expand the RHS Get the variable d all by itself
= d
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Equations
SES ( EQ THE UA EN
Solve these equations without using the distributive law: a
2^ a + 4h = 12
b
..../
.....
7 (n - 3) = 14
2^ a + 4h ' 2 = 12 ' 2 = 6
-
= 6 `a =
c
`n =
- (d + 3) = 11
d
3 (2y + 1) = 18
`y =
`d = e
- (4r + 1) = 19
f
1 (x + 3) = - 2 2
`x =
`r = g
h
- 2 (6 - k) = 5
4 (5 - 2w) = 4 1 2
`w =
`k =
20
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/20..
.
WITH P ONS AR TI
1
SES ( EQ THE UA EN
WITH P ONS AR TI
Equations with parentheses
Where does it work?
Your Turn
Equations
Equations with parentheses 2
Solve these equations after first expanding using the distributive law: a
2^ b - 3h = 8
b 3^w + 1h = 18
#
2 (b - 3) = 8 2b - 6 = 8 2b - 6 + = 8+ 2b '
= 14 ' b =
c
`w =
- ^ q + 2h = 3
d
- ^ x - 6h = - 10
`q = e
`x =
- 5^t + 2h = 25
f
2^3m + 5h = 1
`t = g
`m =
- 2^8 + 5ph = - 24
h
1 ^4a + 8h = - 9 2
`p =
`a =
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Where does it work?
Equations
Combo time: Multi-step equations These require at least two or more steps to solve. Solve these equations by isolating the variable using inverse operations. (i) 6 + 11d = 7 4
6 + 11d # 4 = 7 # 4 4 6 + 11d = 28
d is still not by itself
6 + 11d - 6 = 28 - 6
Subtract 6 from both sides
Multiply both side by 4
11d = 22
d is still not by itself
11d ' 11 = 22 ' 11
Divide both sides by 11
` d = 2 (ii) 4x + 10 = x 4 6
Multiply each numerator by the opposite denominator if both sides are fractions.
4x + 10 = x 4 6 6 # ^4x + 10h = 4 # x
“Cross multiply” the denominators
#
6^4x + 10h = 4x
Expand the left-hand side
24x + 60 = 4x
24x - 24x + 60 = 4x - 24x
Subtract 24x from both sides
60 = - 20x 60 ' (- 20) = - 20x ' (- 20)
Divide both sides by -20
` -3 = x (iii) 3^2 - 3mh = 5^ m - 6h
#
#
3^2 - 3mh = 5^ m - 6h
Expand both sides
6 - 9m = 5m - 30
6 - 9m - 5m = 5m - 5m - 30
Subtract 5m from both sides
6 - 14m = - 30 6 - 6 - 14m = - 30 - 6
Subtract 6 from both sides
- 14m = - 36 - 14m ' (- 14) = - 36 ' (- 14) ` m = 24 7
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Divide both sides by -14
Equations MULT
3
C OM B O TI M E:
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:
`x =
ONS
I-
2
S
1
7 + 5y = 16 2
EP E QUAT I
C OM B O TI ME
Use opposite operations to solve these multi-step equations.
TION
Combo time: Multi-step equations
2x - 5 = - 9 3
I-ST
MULT
Your Turn
STEP EQUA
Where does it work?
`y =
5b + 1 = 2b 7 3
4
3a - 2 = 4a 5 6
`a =
`b =
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Where does it work?
Your Turn
Equations
Combo time: Multi-step equations 5
15h = 14h + 11 4 3
6
3^2 + qh = 6^4 + qh
`q =
`h = 7
3^4n + 1h = 5^4 - nh
8
- 8^ k - 2h = 6^ k + 2h
`k =
`n = 24
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20+
• 1-step equations = 1 point • With variables on both sides = 3 points
2^5a + 12h = 44
AW
*
How many different equations can you make that give the answers below?
..../.....
E SO M
• 2-step equations = 2 points • 3 or more step equations = 5 points
y box? ver
ts scored in poin e
Equations challenge!
Scoring:
Equations
/20...
E *
a= 2
= 5 points!
n= 5
x= 9
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6x - 81 = - 3x = 3 points!
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Equations
Word problems Word problems can be changed into equations to help solve. Write matching algebraic equations for these statements: Unless you are given a variable to use, you can choose any letter to represent a number. (i) A number added to 12 equals 21. Let the number be n ` 12 + n = 21
Pick a letter to represent the number Write the equation using the chosen variable
(ii) 19 subtracted from double a number leaves 23. Let the number be x ` 2x - 19 = 23
Pick a letter to represent the number Write the equation using the chosen variables
The order that the numbers in a subtraction or division are written is important. (iii) Jennifer walks d meters to the shops to meet her friends. Kylie walks an extra 120 m to meet her there. Together, they walk a total of 425 m to get the shops. Write an equation to represent this. The distance walked by Jennifer = d m
Use the letter requested to represent Jennifer
` Kylie walks d + 120 Jennifer + Kylie = 425 m ` d + d + 120 m = 425 m ` 2d + 120 m = 425 m
Collect like terms to simplify
ALWAYS finish word problems with a final statement that answers the question asked. ` The equation that represents the distance walked by Jennifer and Kylie is: 2d + 120 m = 425 m
(iv) Three consecutive numbers (one after the other), when added together equal 18. Let the first number be a
Pick a letter to represent the first number
` a , a + 1 and a + 2 are consecutive numbers Write the equation by adding them all together ` a + a + 1 + a + 2 = 18 Simplify ` 3a + 3 = 18 For consecutive ODD or EVEN numbers, you add 2 instead of 1 at each step.
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Equations
Word problems 1
Match each of the statements below with the correct equation.
a
Seven more than a number is equal to five..............................
7#n = 5
b
A number divided by seven equals five.....................................
c
Five minus a number equals seven...........................................
d
Five added to a number equals seven.......................................
5- n = 7 7 = 5 n n+ 5 = 7
e
The product of five and a number equals seven.......................
f
The difference between a number and five is seven................
n+ 7 = 5 n- 5 = 7
g
Seven times a number equals five.............................................
n '7 = 5
h
The quotient of seven and a number equals five......................
5n = 7
2
Colour in the dot next to the expressions that match each of these statements: a
Fiona’s ipod is m months older than the one bought by her sister 4 months ago. The age of Fiona’s ipod is given by: m–4
m+4
4- m
4m
Janet types 12 words per minute faster than her friend (f) when chatting on a social website. The total words per minute Janet types compared with her friend is given by: b
12 - 2f
12 - f
f + 12
f - 12
Co Tin is one third the age of his father who is x years old. Co Tin’s age is given by: 3 x 3x x+3 x 3 c
d
Catherine runs m meters around a running track with friends Leif and Carol. When Catherine lapped Leif, she was 400 m in front of him and 200 m in front of Carol. Which expression will give the distance ran by all three at that time? m - 600 meter
e
m + 600 meters
3m - 600 meters
3m-200 meters
sold t tickets for a fund raising event. Marcus sold 25 less tickets. Together they sold a total Kyle of 38 tickets. Hint: Ticket sales by Marcus = t - 25
25 + 2t = 28
38 - 2t = 25
2t - 25 = 38
t + 13 = 25
Jackson is 13 cm shorter than the height of his sister (h). Together their heights are exactly 300 cm. 300 = 2h + 13 300 - 2h = 13 2h - 300 = 13 2h - 13 = 300 f
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Your Turn
Equations
Word problems 3
Use x as the variable to write a simplified equation that represents these problems then solve.
a Find three consecutive numbers that when added together equal 12. The three consecutive numbers are: x , x + 1 , x + 2 x + x + 1 + x + 2 = 12 Simplified equation 3x + 3 = 12 Solve equation for x 3x + 3 - 3 = 12 - 3 3x = 9 3x ' 3 = 9 ' 3 x= 3
` The three consecutive numbers are: 3 , 3 + 1 , 3 + 2 = 3,4 ,5
b
Use solution to find the numbers
Find three consecutive ODD numbers that when added together equal 45. Hint: Three consecutive odd numbers are: x , x + 2 , x + 4
The sum of three consecutive numbers is 78. Find the three numbers. c I.f you use x - 1, x and x + 1 the equation formed is simpler to solve. Try it!
d
The sum of four consecutive EVEN numbers is 44. Find the four numbers.
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WOR D P ROBL EMS
WORD PROB LEMS
WORD PROB LEMS
WOR D P ROBL EMS
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Equations
Word problems 4
Write equations for the following problems and then solve.
a Xian thinks of a number n. After multiplying the number by 6 and then adding 2, the answer is 26. What number is Xian thinking of?
b
6n + 2 6n + 2 - 2 6n 6n ' 6 n
= = = = =
26 26 - 2 24 24 ' 6 4
Therefore Xian is thinking of the number 4
Problem as an equation Solve equation for n
Answer with a statement
Freddy thinks of a number also. After multiplying the number by 2, he subtracts 11 and the result is 3 times the original number. What number is Freddy thinking of? Let the number be x.
Kim’s pet dog weighs 4.5 kg less than the dog living next door. Together, both dogs weigh a total .of 25 kg. How much does Kim’s dog weigh? Hint: Let the weight of the dog next door equal d kg c
A number plus one is multiplied by three. This is equal to seven times a number minus five. What is the value of the number? Let the number be n. d
Rememb
e
er me?
A number minus five is multiplied by seven and divided by two. This equals the number plus two all multiplied by three. What is the number?
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Equations
Measurement problems Simplify and solve the following: (i) Find the value of m in this triangle:
3m
+ sum = 180o
7m 3m + 5m + 7m 15m 15m ' 15 `m
= = = =
5m
180 o 180 o 180 o ' 15 12
Angles in a triangle add up to 180o Simplify sum of the angles Solve equation for m
The size of each angle can be calculated by substituting the value for m back into each angle expression. ` 3m = 3 # 12 = 36
36o
5m = 5 # 12 = 60
84o
60o
7m = 7 # 12 = 84
36 o + 60o + 84o = 180 o
(ii) The perimeter of the triangle below is 73 units. Calculate the value of b.
3b
2b + 5
2b + 5 3b + 2b + 5 + 2b + 5 = 73
Perimeter = sum of all the side lengths Simplify by collecting like terms
7b + 10 = 73 7b + 10 - 10 = 73 - 10
Solve equation for b
7b = 63 7b ' 7 = 63 ' 7 `b = 9
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Equations
Measurement problems 1 Write an equation and calculate the value of the variable in each of the diagrams below: b
2c
7c
4b
4c
11b
9c Hint: Angles add to 360o
a
Perimeter = 73 units 3x
b
Perimeter = 90 units 6d
....
/...
../2
0...
BLEMS PRO
Find the value of the variable in each of the diagrams below:
BLEMS PRO
2d + 3
5x - 4 2x + 11 5d + 4
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2
MEASUREME N
T
T
a
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Equations
Formulae Formulae show how different measurements come together for special calculations. Use substitution to calculate the values represented by the following formulae. (i) T he formula V = IR calculates the voltage in a circuit where V is the voltage (in volts), I is the current (in amperes) and R is the resistance of the circuit (in Ohms). Calculate the voltage V if the current I is 25 amperes and the resistance R is 12 ohms. When I = 25 and R = 12 , V = 25 # 12 V Volts = 300
Substitute values into the formula
(ii) T he length of the longest side of a right-angled triangle (c) is found using c = a2 + b2 where a and b are the lengths of the other shorter sides. Calculate the length of the longest side of a right-angled triangle with short sides of length a = 5 cm and b =12 cm.
When a = 5 cm and b =12 cm,
c c c c
= 52 + 122 = 25 + 144 = 169 = 13
After substitution, we are sometimes left with an equation solve. Substitute the given values into these formulae and solve the equation for the unknown variable. (i) T he perimeter of a rectangle (P) is calculated using the formula P = 2l + 2b where l represents the length of the rectangle and b represents the breadth. Calculate the breadth of the rectangle with a perimeter (P) of 34 cm and length (l) of 11 cm. When P = 34 and l = 11 , 34 34 - 22 12 12 ' 2 6
= = = = =
2 # 11 + 2b cm 22 - 22 + 2b 2b 2b ' 2 b
` The breadth of the rectangle is 6 cm
Substitute values into the formula Solve the equation to find b
Answer question with a statement
(ii) T he area of a triangle is found using the formula A = bh 2 where b represents the length of the base side and h represents the perpendicular height of the vertex opposite the base side. What is the height of a triangle with a base length of 12 cm and area of 45 cm2? When A = 45 cm and b =12 cm, 45 = 12h cm2 Substitute values into the formula 2 Solve the equation to find b 45 # 2 = 12h # 2 2 90 = 12h 90 ' 12 = 12h ' 12 7.5 = h ` The breadth of the rectangle is 7.5 cm 32
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Your Turn
Equations
F
0 10 20 30 40
E
50
(i) D = 140 km and T = 2 hours
` The speed S = b
FO
a The speed (S) of an object in kilometers per hour is given by the formula S = . D is the distance in kilometers and T the time taken in hours. Calculate S when: D T
FO
.../20.
..
FO
RM
U LA
..../..
E
Calculate the value of these formulas using the given values for each variable.
ULA
1
130 14
160 50 01
Formulae
7
60
E ULA RM 170 180
AE MUL OR 0 80 90 100 110120
RM
What else can you do?
(ii) D = 31.5 km and T = 0.5 hours
` The speed S =
km/h
km/h
To convert the temperature from degrees Celcius (C) to degrees Fahrenheit (F) the formula is: F = 9C + 160 . Calculate F when: 5 (i) C = 32o (ii) C = 0o
` The converted temperature =
o
F
` The converted temperature =
o
F
2 Substitute the given values into these formulae and solve the equation for the unknown variable.
a
The area of a rectangle is calculated using the formula A=lb where l and b are the length and breadth. Calculate the length l of the following rectangles with these breadths and areas:
(i) A = 36 cm2 and b = 4 cm
(ii) A = 25.48 cm2 and b = 2.6 cm
` The length l of the rectangle = cm b
` The length l of the rectangle = cm
(a + b) , where a and b are 2 the two numbers. Calculate the number a for following average values x and given number b.
The average of two numbers ( x ) is calculated by the formula x = (i) x = 21 and b = 28
(ii) x = 4.75 and b = 3.6
` The number a =
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Your Turn
Equations
Formulae 2
Calculate the value of these formulas using the given values for each variable.
c The speed (S) of an object in kilometers per hour is given by the formula S = D T . D is the distance in kilometers and T the time taken in hours. Calculate T when:
(i) D = 35 km and S = 7 km/h (ii) D = 300 km and S = 60 km/h
` The Time T taken = d
` The Time T taken =
hours
hours
To convert the temperature from degrees Celcius (C) to degrees Fahrenheit (F) the formula is: F = 9C + 160 . Calculate C when: 5 (i) F = 140 o (ii) F = 81.5o
` The converted temperature =
o
C
` The converted temperature =
o
C
e
formula for the area of a trapezium is: A = 2h (a + b) where h is the distance (height) between The the parallel sides and a and b are the lengths of each parallel side. Calculate the height h when:
(i) A = 44 cm2, a = 4 and b = 7 cm
(ii) A = 25.48 , cm2 a = 8 and b = 15 cm
` The height h = cm 34
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` The height h = cm Equations
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Equations
Reflection Time Reflecting on the work covered within this booklet: 1 What useful skills have you gained by learning about equations?
2
Write about one way you think you could apply equations to a real life situations.
3
I f you discovered or learnt about any shortcuts to help with equations or some other cool facts, jot them down here:
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Cheat Sheet
Equations
Here is a summary of the things you need to remember for Equations
Keeping things in balance An equation is an algebraic expression where the left-hand side equals the right-hand side. Left-hand side 0
1
2
3
4
= 5
6
7
8
9
Right-hand side
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Opposite operations Pairs of mathematical operations that do the opposite to each other. and are opposite operations
and are opposite operations Solving simple equations
Solving an equation means: “What value does the variable need to be to keep the equation in balance?” The aim is always to get the variable by itself on one side of the equation using opposite operations. Solutions check Always ask yourself: Does my solution ‘feel’ correct? The best way to find out is by checking your answer to see. Equations with variables on both sides The aim is to get all the variables on one side and the numbers on the other side of the equation. Equations with fractions If one fraction on either side of the equation, cross multiply the denominators. Equations with parentheses Two methods: 1. Divide both sides by the number out the front first.
2. Use the distributive law to expand the parentheses first.
Word problems Change word problems into equations to help solve. Give one of the unknown values a variable. Formulae Use substitution to calculate the value of the formula. Sometimes an equation remains after substitution. This can then be solved as usual. 36
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