yr 9 equations_ and inequalities teacher
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Equa
tion
s&
Ineq
uali
ties
& INEQUALITIESEQUATIONS
SOLUTIONS
www.mathletics.com.au
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Solutions Basics
a x + 3 = 6 is a linear / quadratic equation
(the highest power of x is 1 so this is a linear equation)
b 3x2 = 12 is a linear / quadratic equation
(the highest power of x is not 1 (it’s 2) so this is not a linear equation)
c 3x = 12 is a linear / quadratic equation
(the highest power of x is 1 so this is a linear equation)
d x25 = 3 is a linear / quadratic equation
(the highest power of x is 1 so this is a linear equation)
2. Solve these linear equations:
a 4 7
4 4 7 4
x
x x
x 3
+ =
+ = +
=
b 3 10
3 3 10 3
x
x
x 7
+ =
+ =
=
c 2 13
2 2 13 2
x
x
x 15
=
+ = +
=
d 6 10
6 6 10 6
x
x
x 16
=
+ = +
=
Page 4 questions
1. Choose whether the following equations are linear or quadratic by circling the appropriate word (don’t solve):
Subtract 4 from both sides
Subtract 3 from both sides
Add 6 to both sides
Add 2 to both sides
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Page 5 questions
3. Find the value of the variable in each of these linear equations:
a x
x
x
2 8
22
28
4
=
=
=
c x
x
x
45
44 5 4
20
# #
=
=
=
e x
x
x
x
x
23 9
23 2 9 2
3 18
33
318
6
# #
=
=
=
=
=
b x
x
x
5 35
55
535
7
=
=
=
d x
x
x
38
33 8 3
24
# #
=
=
=
f x
x
x
x
x
35 5
35 3 5 3
5 15
55
515
3
# #
=
=
=
=
=
Divide both sides by 2
Multiply both sides by 4
Multiply both sides by 2 Multiply both sides by 3
Divide both sides by 3 Divide both sides by 5
Multiply both sides by 3
Divide both sides by 5
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Solutions Basics
a
4 20
x
x
x
x
x
4 3 23
4 3 23 3
4 20
4 4
5
+ =
=
=
=
=
c m
m
m
m
m
5 6 31
5 6 6 31 6
5 25
55
525
5
+ =
+ =
=
=
=
e k
k
k
2 10
22
210
5
=
=
=
b a
a
a
a
a
2 5 9
2 5 5 9 5
2 14
22
214
7
=
+ = +
=
=
=
d n
n
n
n
n
4 7 24
4 4 7 24 4
7 28
77
728
4
+ =
+ + = +
=
=
=
f Hint( : )m mm
m
m
m
m
14 6
4 4 6 4
2
1
1
12
2
=+ =
+ =
=
=
=
Page 6 questions
4. Solve for the variable in each linear equation:
Subtract 3 from both sides
Subtract 6 from both sides Add 4 to both sides
Subtract 4 from both sides
Add 5 to both sides
Divide both sides by 4
Divide both sides by 5
Divide both sides by –2
Divide both sides by 7
Divide both sides by –1
Divide both sides by 2
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a 4
4 4 4
y
y
y
4
4
16
# #
=
=
=
c
2 54
p
p
p
p
p
9
26
9
29 6 9
2 54
2 2
27
# #
=
=
=
=
=
e 14
3 14 3
7 42
x
x
x
x
x
37
37
77
742
6
# #
=
=
=
=
=
b 15
4 15 4
5 60
x
x
x
x
x
45
45
55
560
12
# #
=
=
=
=
=
d
4 88
d
d
d
d
d
114 8
114 11 8 11
4 88
4 4
22
# #
=
=
=
=
=
f
10 100
m
m
m
m
m
410 25
410 4 25 4
10 100
10 10
10
# #
=
=
=
=
=
a
so or
y
y
y
y
y
y y
9 0
9 9 0 9
9
3
3
3 3
2
2
2
2 2!
!
=
+ = +
=
=
=
= =
^ h
b
so or
4 100
25
5
5 5
x
x
x
x
x
x x
44
4100
5
2
2
2
2 2!
!
=
=
=
=
=
= =
^ h
Page 7 questions
5. Solve for the variable in each linear equation:
6. Solve these quadratic equations:
Multiply both sides by 4
Multiply both sides by 9
Multiply both sides by 3
Add 9 to both sides
Since –32 = 9 and 32 = 9
Find the square root of both sides
Multiply both sides by 11
Multiply both sides by 4
Divide both sides by 2
Divide both sides by 7
Divide both sides by 4
Divide both sides by 10
Divide both sides by 4
Find the square root of both sides
Multiply both sides by 4
Divide both sides by 5
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Solutions Knowing More
a
1
1
110
u u
u u u u
uu
u
2 10 3
2 3 10 3 3
1 10 10 0 10
10
=
=
+ = +
=
=
b 7 18 3 10
7 3 18 3 3 10
4 18 18 10 18
4 28
x x
x x x x
x
xx
x44
428
7
= +
= +
+ = +
=
=
=
c x
x
x
x
x
x
3 2 3
3 6 3
3 6 6 3 6
3 9
33
3
9
3
+ =
+ =
+ =
=
=
=
^ h
d 4 18 12 2
4 2 18 12 2 2
6 18 18 12 18
6 6
y y
y y y y
y
y
y 1
+ =
+ + = +
+ =
=
=
e 10 2
10 60 20 2
10 2 60 20 2 2
8 60 20
8 60 60 20 60
8 80
n n
n n
n n n n
n
n
nn
n
6 10
88
880
10
= +
= +
= +
=
+ = +
=
=
=
^ ^h h
Page 11 questions
1. In these linear equations the variable appears on boths sides. Solve for the missing value:
Subtract 3u from both sides
Subtract 3x from both sides
Add 10 to both sides
Add 18 to both sides
Divide both sides by -1
Divide both sides by 4
Expand the brackets
Subtract 6 from both sides
Divide both sides by 3
Subtract 18 from both sides
Divide both sides by 6
Expand the brackets
Add 60 to both sides
Divide both sides by 8
Subtract 2n from both sides
Add 2y to both sides
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f 6 4 5
6 4 5 15
6 5 4 5 5 15
11 4 15
11 4 4 15 4
11 11
m m
m m
m m m m
m
m
mm
m
3
1111
1111
1
= +
=
+ = +
=
+ = +
=
=
=
^ h
g 8 5 3 4
8 32 5 3 4
3 29 4
3 29 29 4 29
3 33
k k
k k
k
k
kk
k
4
33
333
11
+ =
+ =
=
+ = +
=
=
=
^ h
h 5 6 3 6
10 5 6 12 3 6
4 10 6
4 10 10 6 10
4 4
y y
y y
y
y
yy
y
2 1 2
44
44
1
+ =
+ + =
+ =
+ =
=
=
=
^ ^h h
Page 11 questions
1. In these linear equations the variable appears on boths sides. Solve for the missing value:
Expand the brackets
Add 5m to both sides
Add 4 to both sides
Divide both sides by 11
Expand the brackets
Add 29 to both sides
Divide both sides by 3
Collect like terms
Expand the brackets
Subtract 10 from both sides
Divide both sides by 4
Collect like terms
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i 8 12
8 2 18 12
6 18 12
6 18 18 12 18
6 30
t t
t t
t
t
t
t
t
2 18
66
6
30
5
=
+ =
+ =
+ =
=
=
=
^ h
j 2 3 10
2 6 3 12 10
6 10
6 6 10 6
4
a a
a a
a
a
aa
a
3 4
1 1
4
4
+ + =
+ =
=
+ = +
=
=
=
^ ^h h
Page 11 questions
1. In these linear equations the variable appears on boths sides. Solve for the missing value:
2. Find Ivan's mistake when he tried to solve this equation?
3 2 2 72 2 2 2
h h
h h
h h
h h
h
h h
3 2 2 1 5
3 2 2 2 5
3 2 2 7
5
+ = + +
+ = + +
+ = +
+ = +
=
^ ^h h
Subtract 18 from both sides
Divide both sides by 6
Expand the brackets
Expand the brackets
Oops - Expanded the brackets incorrectly
should be 3h + 6 = 2h + 2 + 5
Correct answer then comes out to h = 1
Add 6 to both sides
Divide both sides by -1
Collect like terms
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a 1 4
1 1 4 1
5
8 5 8
x
x
x
x
x
8
8
8
840
# #
=
+ = +
=
=
=
b n n
n n
n n
n n
n n n n
n
24
125 5
122
4 12125 5
212 48
1260 60
6 48 5 60
6 5 48 48 5 5 60 48
12
# #
+ = +
+ = +
+ = +
+ = +
+ = +
=
` `j j
c b
b
b
b
b
b
53 4 4
3 4 5 4
3 4 20
3 4 4 20 4
3 24
8
#
+ =
+ =
+ =
+ =
=
=
Add 1 to both sides
Multiply both sides by 8
Multiply both sides by denominator 5
Lowerst common Denominator (LCD) is 12
Multiply both sides by (LCD)
Expand brackets
Page 12 questions
3. Solve these linear equations which contain fractions:
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d c c
c c
c c
c
c
c
212
22
2 12
2 24
3 24
33
324
8
# #
+ =
+ =
+ =
=
=
=
` j
e r
r
r
r
r
r
r
516 2 10
16 2 10 5
16 2 50
16 2 2 50 2
16 48
1616
1648
3
#
+ =
+ =
+ =
+ =
=
=
=
f m m
m m
m m
m m
m
m
m
2 35
62 3
6 5
26
36 30
3 2 30
5 30
55
530
6
# #
+ =
+ =
+ =
+ =
=
=
=
` j
Page 12 questions
3. Solve these linear equations which contain fractions:
Multiply both sides by LCD (2)
Multiply both sides by 5
Multiply both sides by LCD
Divide both sides by 16
Divide both sides by 5
Divide both sides by 3
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a q q
q q
q q
q q
q q q
q
q
q
3
7 5
6
4 30
63
7 56
6
4 30
2 7 5 4 30
14 10 4 30
14 4 10 10 4 30 10
10 40
1040
4
# #
#
+=
+=
+ =
+ =
+ =
=
=
=
c c
^
m m
h
b
u u
u u
u u
u u
u
u
u
u
u
65 5
95 10 2 7
186
5 5 189
5 10 18 2 18 7
3 5 5 2 5 10 36 126
15 15 10 20 36 126
25 71 126
25 71 71 126 71
25 55
2525
2555
511
# # # #
# #
+ + + + =
+ + + + =
+ + + + =
+ + + + =
+ =
+ =
=
=
=
` `
^ ^
j j
h h
c
g g g
g g g
g g g
g g g
g g g g
g
3
2
10 21
4
3
603
260
1060
21 60
4
3
20 2 6 30 15 3
40 6 30 45
46 45 30 45 45
30
# # # #
# #
+ = +
+ = +
+ = +
+ = +
= +
=
Page 13 questions
4. Solve these linear equations which contain fractions:
LCD is 6
Multiply by LCD
LCD is 60
Multiply by (LCD)
Multiply by LCD
LCD is 18
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d x
xx
x
x
x
x
2 6
2 6
2 6
22
26
3
# #
=
=
=
=
=
e
d
dd
d
d
d
d
d
d
8 26 1
8 28 2
6 1 8 2
6 8 2
6 2 8 2 2
8 8
88
88
1
# #
=
=
=
+ = +
=
=
=
^ ^h h
f 2
2
3 45 2
3 2 45 2 2 45
k
k
kk
kk
k k
k k k k
k
3 45
3 45
45
# #
=
=
=
+ = +
=
Page 13 questions
Multiply both sides by the denominator
Multiply both sides by the denominator
You could rewrite this as 8d - 2 = 6 if you prefer
Multiply both sides by the denominator
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Page 14 questions
5. Three times a number is 45. What is the number?
Let the number be n
3 45nn
n33
345
15
=
=
=
6. Claire, Leanne and Lindsay are sisters. Claire is two years older than Leanne and Leanne is 4 years older than Lindsay. The sum of all their ages is 54. How old is each sister?
Let Leanne's age be x
Ages: Claire + Leanne + Lindsay = 54
54
3 2 54
3 2 2 54 2
3 56
x x x
x
x
xx
x
2 4
33
356
1832
+ + + =
=
+ = +
=
=
=
^ ^h h
So Leanne is 1832 , Claire is 20
32 and Lindsay is 14
32
7. Charlie has been collecting stamps which he keeps in two separate books. The second book has 7 more than triple the stamps of the first book. If he has 35 stamps in total (from both books) then:
Let the number of stamps in the first book be xThe number of stamps in the second book is 3x + 7
Book book total stamps1 2
3 7 35
4 7 35
4 7 7 35
4 7 7 35 7
4 28
x x
x
x
x
xx
x44
428
7
+ =
+ + =
+ =
+ =
+ =
=
=
=
a How many stamps are in the first book? 7
b How many stamps are in the second book? 35 - 7 = 28
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Page 14 questions
8. Victor has a bag filled with 2c and 5c coins. He has 2 more coins worth 2c than the coins worth 5c. How many 2c coins does Victor have if all his coins sum to 88c?
Let n be the number of 5c coins. He has (n + 2) of 2c coins.
They all add up to 88c so,
2 5 88
2 4 5 88
7 4 88
7 4 4 88 4
7 84
n n
n n
n
n
nn
n
2
77
784
12
# #+ + =
+ + =
+ =
+ =
=
=
=
^ h
So he has 12 coins worth of 5c and 14 coins worth of 2c.
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2. Solve these inequalities:
a
c
b
d
3 4
3 3 4
x
x
x 1
1
1
1
+
+
4 5
4 4 5 4
x
x
x 9
$
$
$
+ +
m
m
m
7 4
7 7 4 7
11
$
$
$
+
+
10 8
10 10 8 10
p
p
p 2
#
#
#
+ +
Page 18 questions
1. Identify if the following are true or false:
a 6 32 True
b 5 81 True
c 3 82 False
d 2 51 False
e 4 42 False
f 8 41 True
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Page 18 questions
2. Solve these inequalities:
e
g
i
f
h
j
q
q
q
5 35
5
5
535
7
#
#
#
12
h
h
h
h
h
4 3 51
4 3 3 51 3
4 48
44
448
2
2
2
2
2
+
+
x x
x x x x
x
x
x
5 24
5 24
4 24
44
424
6
2
2
2
2
2
+
+
x x
x x
x x x x
x
x
3 1 3 1
3 3 3 3
3 3 3 3 3 3 3 3
6 6
1
#
#
#
#
#
+ + + +
^ ^h h
h
h
h
h
h
h
h
2
4 82
22
4 82 2
4 8 4
4 8 8 4 8
4 4
44
44
1
# #
1
1
1
1
1
1
1
+ +
c m
14
20 20 14 20
5 3 4 2 280
15 8 280
7 280
x x
x x
x x
x x
x
x
x
43
52
43
52
77
7280
40
# # #
# #
$
$
$
$
$
$
$
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Page 19 questions
3. Graph these inequalities:
a x 32
c x 0#
e x5 51#
g and0 3x x1 $
b
x 2$
d x1 31 1
f x2 41 #
h and4 1x x2 #
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5
4. Write down the inequality represented by each of the following graphs:
-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5
a b
c d
e f
x 41 4x #
1x $- x2 41#-
x4 01 #- 3x #-
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Page 20 questions
5. Solve these more complicated linear inequalities, then graph their solution:
a 5 6 9
5 6 9 18
5 9 6 6 18 6
4 12
x x
x x
x x
xx
x
2
4
4
412
3
2
2
2
2
1
1
+ +
+ +
+
^ h
Reverse the inequality when dividing by a negative number
c 2
8 8 2
4 3 16
4 4 3 16 4
3 12
4
d
d
d
d
d
d
d
d
8
4 3
8
4 3
3
3
312
4
# #
#
#
#
#
#
#
#
$
c m
Reverse the inequality when multiplying by a negative number
b 7 4 5
7 4 16 5
3 16 16 5 16
3 21
y y
y y
y
y
y
y
4
33
321
7
2
2
2
2
2
2
+
+ +
^ h
-2 -1 0 1 2 3 4 5 6 7 8-5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5
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Page 20 questions
5. Solve these more complicated linear inequalities, then graph their solution:
d b b
b b
b b
b
b
b
b
8 35
248 3
5 24
3 8 120
5 120
55
5120
24
24
# #
#
#
#
#
#
#
$
` j
Multiply both sides by LCM of all fractions (24)
Reverse the inequality when multiplying by a negative number.
f
.
c c
c c
c c
c c c c
c
c
c
48
6 2
21
12
4848
6 248
21 48
12
6 2 24 4
6 4 2 2 24 4 4 2
10 26
1010
1026
2 6
# # #
1
1
1
1
1
1
1
+ + + +
c m
Multiply both sides by LCM of all fractions (48)
Reverse the inequality when multiplying by a negative number
e a a
a a
a a
a a a a
a
a
a
a
2 104 3
102
1010
4 3
5 4 3
5 4 4 4 3
9 3
99
93
31
31
# #
$
$
$
$
$
$
$
#
+
+
+
+
` `j j
Multiply both sides by LCM of all fractions (10)
Reverse the inequality when multiplying by a negative number22 23 24 25 26 27 28 29 30 31 32
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5
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Page 26 questions
1. Write down 2 possible solutions for the variables in these equations:
a x + y = 4
Anything where x and y add up to 4
Eg x = 3, y = 1 or x = 2, y = 2 or x = 4, y = 0 or x = 5, y = -1 or x = 3.5, y = 0.5 etc
Can also write these pairs like this: (3, 1), (2, 2) (4, 0), (5,-1), (3.5, 0.5)
b 2a + b = 6
Anything where twice a, plus b, makes 6.
Eg a = 1, b = 4, or a = 2, b = 2 or a = 3, b = 0 or a = 4, b = -2 or a = 0.5, b = 5 etc
c 3x - 4y = 10
Anything where 3 times the first number, minus three times the second number, makes 10.
Eg x = 6, y = 2, or x = -2, y = -4 or x = 30, y = 20 or x = 2, y = -1 or x = 1, y = -1.75 etc
2. Solve for the variables in these simultaneous equations using the substitution method:
a 2x + y = -1 1
x - 2y = -4 2
Make one of the variables the subject of 1
y = -1-2x
Substitute this into 2
2 4
2 4 4
5 2 4
5 6
.
x x
x x
x
x
x
1 2
56 1 2
=
+ + =
+ =
=
= =
^ h
Substitute this x value into either equation, say 2
2 4
1.2 2 4
2 4 1.2
.
.
x y
y
y
y
y2
2 8
1 4
=
=
= +
=
=
so x = -1.2 and y = 1.4
b 2p + 3q = 10 1
2q - 4p = 44 2
Make one of the variables the subject of 2 , say q
2 44 4q pq p
q p2
2
244
2
4
22 2
= +
= +
= +
Substitute this into 1
2 3 10
2 66 6 10
8 66 66 10 66
8 56
p p
p p
p
p
p
22 2
7
+ + =
+ + =
+ =
=
=
^ h
Substitute p back into either equation to solve for q q
q
q
q
q
2 7 3 10
14 3 10
3 14 14 10 14
3 24
8
# + =
+ =
+ = +
=
= so p = -7 and q = 8
1
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3. Use the graphical method to solve for these equations:
-2 0 2 4
4
2
-2
-4
-6
y
x
987654321
-1-2
-7 -6 -5 -4 -3 -2 -1 0 1 2
y
x
a 3x + 2y = 2 1
2x - y = 6 2
Make y the subject of both equations
y x
y x
2
31
2 6
= +
=
The lines intersect at (2,-2) so x = 2 and y = -2
b 3y - 4x = 24 1
2y + 2x = 2 2
Make y the subject of both equations
y x
y x34 8
1
= +
= +
The lines intersect at (-3, 4) so x = -3 and y = 4
2
1
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4. Solve for the variables in these simultaneous equations using the elimination method:
a 3x - y = -15 1
y + 2x = 0 2
In this case we can eliminate y by adding the two equations together.
x y
y x
x x
x
x
3 15
2 0
3 2 15
5 15
3
=
+ =
+ =
=
=
1
2
1 + 2
Substitute x = -3 into either equation – say 1 .
3 3 15
9 15
y
y
y 6
# =
=
=
so x = -3 and y = 6
b b - 4a = -12 1
3a - 2b = -1 2
Make the coefficient of b the same in both equations; then subtract 2 from 1
2b - 8a = -24 1 # 2 3a - 2b = -1 2
5 25a
a
a
55
525
5
- =-
-- =
--
=
(2 # 1 ) + 2
Substitute a = 5 into either equation, say 2
3 5 2 1
15 2 1
2 15 15 1 15
2 16
b
b
b
b
b 8
# =
=
+ =
=
=
So a = 5 and b = 8
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a 7m = 16 - 6n
2n = 3m + 16
Page 29 questions
5. Solve these simultaneous equations using any method:
Page 30 questions
6. Solve these simultaneous equations using any method:
a 8c - 8d = 2
9c - 8d = 5
1
1
1
2
2
23c
c 3`
- =-
=
-
substitute c 3= into 1 to find d:
Substitute this into 1 :
Substitute this into 1 to solve for n:
d
d
d
8 3 8 2
8 22
822
411
`
`
`
- =
=
= =
^ h
and3c d411` = =
In 2 , make n the subject (find n by itself):
n m2
3 16= +
m m
m m
7 16 62
3 16
7 16 9 48
`
`
= - +
= - -
` j
2 3 16n
n
2
5
= - +
=
^ h
16 32
2
m
m
`
`
=-
=-
Now, solve for m:
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7. Solve these simultaneous equations using any method:
a 6a - 2b - 10 = 0
2a + 3b - 29 = 0
1
1
2
2
Multiply 2 by 1 and subtract:
Substitute this into 2 :
6 2 10 0a b
a b6 9 87 0
- - =
+ - =
11 77
7
b
b`
- =-
=
2 3 2 0
2 8
a
a
a
7 9
4
+ - =
=
=
^ h
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8. Find equations and solve them for these word problems (using any method):
a The sum of two numbers is 12. The sum of the first number, and double the second number is 16. What are the numbers?
Call the numbers a and b
a + b = 12 1 (The sum of two numbers is 12)
a + 2b = 16 2 (The sum of the first number, and double the second number is 16)
You can solve this using any of the 3 methods, for example elimination. We can choose to eliminate a since it already has the same coefficient in both equations.
b - 2b = 12 - 16 1 - 2
-b = -4
b = 4
Substitute b = 4 into 1
a + 4 = 12
a = 8
So a = 8 and b = 4
b Ari is three years older than Eric. In three years from now, Ari will be twice as old as Eric will be. How old are
they now?
Lets call Eric’s present age, e, and Ari’s present age a
a = e + 3 1
3a e23+ = + 2
You can solve this using any of the 3 methods, for example substitution.
Substitute 1 into 2
e e
e e
e e
e
23 3 3
226 2 3
6 2 6
0
# #
+ + = +
+ = +
+ = +
=
` ^j h
Substitute e = 0 into either equation gives a = 3
So Eric is 0; Ari is 3.
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8. Find equations and solve them for these word problems (using any method):
c A resturaunt sells two kinds of meals: pizza and pasta. A pizza costs $14 and a pasta costs $10. In a single day the resturaunt sold 79 meals. If they earned $994 on this day, how many of each meal was sold?
If we call number of pizzas sold is x and number of pasta meals sold is y, then x+ y = 79 (79 meals sold in total)
14x + 10y = 994 ($14 for every pizza + $10 for every pasta meal = $994 in total) You can solve this with any of the 3 methods, for example graphically
Careful measurement will show that x = 51 and y = 28. So 51 pizzas and 28 pasta meals were sold.
Note: This method is approximate, and the answer will not always be exact.
y
x -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0-5
-10
-15
-20
-25
-30
-35
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