Transcript Simple Algebraic Operations

Order of operations
Formulae
Removing brackets
Factorising
Equations
Inequations
Algebra Menu
Presents
Hands up!
3+5x2
?
13
A bit of order please!
Brackets
B
Of
O
Multiply
M
Divide
D
Add
A
Subtract
S
( 3 + 4 ) x 3 = 7 x 3 = 21
5 + ½ of 8
=5+4 =9
3+5x2
= 3 + 10 = 13
18 - 12 ÷ 3
= 18 - 4 = 14
These are the very last things you do!
BOMDAS
BOMDAS Practice
(1) 3 + 7 x 6
= 3 + 42 = 45
(2) (3 + 7) x 6
= 10 x 6 = 60
(3) 32 – 24 ÷ 4
= 32 - 6 = 26
(4) 5 + 6 x 3 - 1
= 5 + 18 – 1 = 22
(5) 5 + 6 x (3 – 1) = 5 + 6 x 2 = 5 + 12 = 17
Algebra Menu
Presents
Algebra Short hand
3a means
bc means
x² means
3 times a
b times c
x times x
m/k means
m divided by k
e means
f
e divided by f
Formulae (1)
Given that C
= s + nr, calculate C when s=12, n=5 and r=8.
C = s + nr
C = 12 +5
5x8
C = 12 + 40 = 52
M
BOMDAS
Formulae (2)
d
Given that p = as +
m
calculate p when a=5, s=9, d=18 and m=6.
d
p = as +
m
18
p=5x9 +
6
p = 45 + 3 = 48
BOMDAS
MD
Formulae (3)
Given that s
= 3t² - 6t + 5 calculate s when t=4.
s = 3t² - 6t + 5
s= 3x4x4 - 6x4+5
s = 48 - 24 + 5 = 29
BOMDAS
M
Now do Exercise 1
(HSDU Support Materials))
Algebra Menu
Presents
Removing brackets (1)
b + 5) = 3b
3(b
3
This means 3 times b
Removing brackets (1)
3(b + 5
5) = 3b + 15
3
This means 3 times 5
Removing brackets (2)
5
8 - f) = 40
5(8
Removing brackets (2)
5
5(8 - ff) = 40 - 5f
Tip: Draw lines to keep you right.
5(8 – f) = 40 - 5f
Algebra Menu
Now do Exercise 2
Question 1
(HSDU Support Materials))
Removing brackets (3)
3b + 2) = 12b
4(3b
4
4 x 3b = 4 x 3 x b = 12 x b = 12b
Removing brackets (3)
4(3b + 2
2) = 12b + 8
4
Removing brackets (4)
3
7a – 2b) =21a
3(7a
Removing brackets (4)
3
3(7a –- 2b
2b) =21a - 6b
Removing brackets (5)
c
3c + b) = 3c²
c(3c
c x 3c = 3c x c = 3c²
Removing brackets (5)
c
c(3c + b
b) = 3c² + bc
Tip: Remember to draw lines to keep you right.
c(3c + b) = 3c² + bc
Removing brackets (6)
6
5e + 3f - 2) = 30e
6(5e
Removing brackets (6)
6
6(5e + 3f - 2) = 30e + 18f
Removing brackets (6)
6
6(5e + 3f - 2
2) = 30e + 18f - 12
Careful!
Tip: Remember to draw lines to keep you right.
6(5e + 3f -2) = 30e + 18f - 12
Algebra Menu
Now do Exercise 2
Questions 2 & 3
(HSDU Support Materials))
Simplifying (1)
4
3a + 5) - 8 = 12a
4(3a
Simplifying (1)
4
4(3a + 5
5) - 8 = 12a + 20 - 8
This is not inside
the brackets
So don’t multiply!
Simplifying (1)
4(3a + 5) - 8 = 12a + 20 - 8
This can be tidied up
= 12a + 12
Simplifying (2)
5b – 6) = 3b + 10b
3b + 2
2(5b
This is not inside
the brackets
So don’t multiply!
Simplifying (2)
6 = 3b + 10b - 12
3b + 2
2(5b – 6)
Careful!
Simplifying (2)
3b + 2(5b – 6) = 3b + 10b - 12
This can be tidied up
= 13b - 12
Algebra Menu
Now do Exercise 2
Questions 4, 5.
(HSDU Support Materials))
Algebra Menu
Presents
Factorising (1)
?
4
4a – 6b
6 = 2
What’s the highest
number that will divide
into both 4 and 6?
This is called the
highest common factor
Factorising (1)
?
4a – 6b = 2( 2a
)
Factorising (1)
?)
4a – 6b = 2( 2a - 3b
Factorising (1)
4a – 6b = 2( 2a - 3b )
Factorising (2)
?
b+b
3b
bc = b
What’s the
highest common factor?
This time it’s the letter
b that they have in
common.
Factorising (2)
3b + bc = b( 3?
)
Factorising (2)
3b + bc = b( 3 + c? )
Factorising (2)
3b + bc = b( 3 + c? )
Factorising (3)
?f
6f + 9gf
6
9 =3
What’s
3 divides
intothe
6 and 9.
highest common factor?
Notice that the letter
f is also common to
both.
Factorising (3)
2
6f + 9gf = 3f( ?
)
Factorising (3)
? )
6f + 9gf = 3f( 2 + 3g
Factorising (3)
6f + 9gf = 3f( 2 + 3g )
Algebra Menu
Now do Exercise 3
(HSDU Support Materials))
Algebra Menu
Presents
x + 5 = 20
x + 5 = 20
x+5
(-5)
20
x + 5 = 20
x
20
(-5)
x + 5 = 20
x
15
x = 15
x + 5 = 20
Example 1:
x + 5 = 20
(-5) (-5)
x = 15
Exercise A
Answers
1) x + 8 = 12
(-8) (-8)
x=4
2) a + 12 = 30
(-12) (-12)
a = 18
3) 5 + x = 13
(-5)
(-5)
x=8
4) b + 7 = 42
(-7) (-7)
b = 35
3x + 7 = 25
3x + 7 = 25
3x+7
(-7)
25
3x + 7 = 25
3x
25
(-7)
3x + 7 = 25
3x
18
3x = 18
?
3x = 18
This means 3 times x equals 18
3 times 6 equals 18
So x must be 6
x = 6
3x + 7 = 25
Example 2:
3x + 7 = 25
(-7) (-7)
3x = 18
x=6
Exercise B
1) 2x + 7 = 23
(-7) (-7)
2x = 16
x=8
3) 5 +8y = 37
(-5)
(-5)
8y = 32
y=4
Answers
2) 6a + 11 = 29
(-11) (-11)
6a = 18
a=3
4) 9 + 3d = 42
(-9)
(-9)
3d = 33
d = 11
5x - 6 = 44
5x - 6 = 44
5x-6
(+6)
44
5x - 6 = 44
44
5x
(+6)
5x - 6 = 44
5x
50
5x = 50
5x - 6 = 44
Example 3:
5x - 6 = 44
(+6) (+6)
5x = 50
x = 10
Exercise C
Answers
1) 5b - 8 = 12
2) 7a - 9 = 33
(+9) (+9)
(+8) (+8)
7a = 42
5b = 20
a=6
b=4
3) 6y - 6 = 30 4) 9z- 7 = 20
(+7) (+7)
(+6) (+6)
9z = 27
6y = 36
z=3
y=6
Is there a
shortcut?
Change side, change sign
5x - 6 = 44
Change side, change sign
5x
= 44 + 6
5x = 50
x = 10
5x - 6 = 44
Example 4:
5x - 6 = 44
5x
= 44 + 6
5x = 50
x = 10
Change side, change sign
3x + 7 = 25
Change side, change sign
3x
= 25 - 7
3x = 18
x = 6
3x + 7 = 25
Example 5:
3x + 7 = 25
3x
= 25 - 7
3x = 18
x=6
Exercise D
1) 2a + 7 = 23
2a = 23 - 7
2a = 16
a=8
2) 7x - 9 = 33
7x = 33 + 9
7x = 42
x=6
Answers
Tricky one!
6x = 9 + 2x
6x – 2x = 9
4x = 9
?
Change side, change sign
4x
4 = 9
The 4 is multiplying the
x, so when it changes
side, it divides.
Change side, change sign
4x
4 = 9
x=9 ÷ 4
x = 2.25
5x + 2 = 3x + 6
5x + 2 = 3x + 6
5x+2
(-2)
3x+6
5x + 2 = 3x + 6
5x
3x+6
(-2)
5x + 2 = 3x + 6
5x
3x+4
(-3x)
5x + 2 = 3x + 6
4
5x
(-3x)
5x + 2 = 3x + 6
2x
2x = 4
4
So x = 2
Change side, change sign
5x + 2
5x
5x
5x –3x
2x
x
=
=
=
=
=
=
3x + 6
3x + 6 - 2
3x + 4
4
4
2
Change side, change sign
3x + 6 = 8x - 4
3x + 6 + 4 = 8x
Most
Xs
are
on
the
right.
3x + 10 = 8x
So this time:
10 = 8x – 3x
Xs on the right,
10 =on5x
numbers
the left!
x = 2
3x + 6 = 8x - 4
Example 6:
3x + 6 = 8x - 4
Try two changes
together!
6+4 = 8x – 3x
10 = 5x
x=2
Exercise E
1) 5a + 7 = 2a + 22
5a – 2a = 22 - 7
3a = 15
a=5
2) 3x + 9 = 7x - 15
9 + 15 = 7x – 3x
24 = 4x
x=6
Answers
Algebra Menu
Presents
Inequations
8
9
9
>
is greater
than
8
Inequations
5
9
5 <
is less
than
9
Remember
is greater than
is less than
Hint: The less than sign
is almost L shaped.
<
ess than
Practice: fill in the correct sign
7
<
10
(2) 12
>9
(3) 2
<
3
(4) 5
>
(5) 3
>
-2
(6) -6
< -3
(7) -2
<
2
(8)
-1
<
0
(9) -4
>
-5
(10) 3
<
3
(1)
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A
B
C
D
is less than
Next question
A
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D
is greater than
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A
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C
D
is less than or equal to
Finish
A
B
C
D
Solving inequations
Use the same
method as for
equations!
3a + 8 < 26 - 8
3a < 18
a < 6
3a + 8 < 26
Example:
3a + 8 < 26
3a
< 26 - 8
3a < 18
a<6
Now do Exercise ?
(HSDU Support Materials))
K Hughes 2003
Aged
greater than
or equal to
18
Aged
greater than
or equal to
17
Aged
greater than
or equal to
16
Speed
less than or
equal to
30 mph
Passengers
less than or
equal to
5
Number Line
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
negative
numbers
Negative
positive
numbers
Positive