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5th year Algebra Revision
Rules for addition and subtraction


Add like signs and keep the sign eg. -4-3=-7
Subtract unlike signs and keep sign of bigger number eg. -9+3=-6, 10-8=2
Rules for Multiplication and Division


Minus*Minus=plus eg.
Minus*plus= minus eg.
-8*-2=+16
-7*3=-21
Removing Brackets
1. Multiply each term inside by whatever is outside the bracket(including signs)
2. Then add and subtract terms that are the same.
Example : Simplify
3x(2x+4y)+6y2-6y(x+y)
6x2+12xy +6y2-6xy-6y2
Add and subtract like terms
6x2+6xy
Multiplying out Brackets
eg. (x+5)(x-3) Can use boxes (array method)
X
X
-3
X2
-15
X2-3x+5x-15
Ans: X2 +2x -15
1st term
0r use
2. Multiply top value by side
value in the table (-3)(5)=15
3. Write everything in the boxes
underneath and tidy up the
same terms. -5x+3x=2x
-3x
5x
+5
1. Put 1 bracket across the top
and the other along the
side.
2nd term
(x+5)(x-3)
2nd bracket
1st term(2nd bracket) +2nd term (2nd bracket)
x (x-3)
X2 - 3x
+
+
5
(x-3)
5x-15
X2 +2x -15 (same answer as above)
Multiplying out Brackets
1. Put 1 bracket across the top
and the other along the
side.
Eg. (5a-2)2
2. Multiply top value by side
value in the table (-2)(-2)=+4
3. Write everything in the boxes
underneath and tidy up the
same terms. -10a-10a=-20a
(5a-2)(5a-2)
5a
5a
-2
25a2
-10a
-10a
+4
-2
25a2-10a-10a+4
25a2-20a+4
1st term
0r use
2nd term
2nd bracket
(5a-2)(5a-2)
1st term(2nd bracket) +2nd term (2nd bracket)
5a (5a-2)
-2
(5a-2)
25a2-10a-10a+4
25a2-20a+4 (same answer as above)
Multiplying terms with Brackets and Simplifying
Eg. -4(x-1)+2(x-8)
-4x + 4+ 2x- 16
Multiply everything inside by what`s inside brackets including signs
Tidy up terms that are the same eg. -4x+2x=-2x, +4-16=-12
-2x-12
Eg. 5(m2-2m-1)-4(m2-m-1)
Multiply out brackets ( - X - = +)
5m2-10m-5-4m2-4m+4
add or subtract like terms
1m2-14m-1
eg. -10m-4m=-14m (2 same signs add and keep sign)
-5+4=-1 (2 different signs subtract and keep sign of bigger num)
5m2-4m2=1m2 (2 different signs subtract and keep sign of bigger
num)
Solving Equations
Follow the steps to solve
1.
2.
3.
4.
Multiply out brackets
Put x`s on the left of the equals & numbers on the right(anything moves sides it changes sign)
Tidy up x`s and tidy up numbers
To get x on its own you ÷ whatever is stuck to it into the other side.
Example:
5x - 3= 17
5x = 17+3
5x = 20
x = 20
5
x`s on left numbers on right move -3 and becomes+3
tidy up numbers-- 2 same signs add and keep sign
to get x ÷ 5 into 20
To get x on its own you ÷ whatever is stuck to it into the other side
x = 4
Example:
4x-2 = 5x -5
-5x+4x = -5+2
-1x
= -3
X = -3
-1
X = 3
x`s on left changes to -5x numbers on right move -2 and becomes+2
2 different signs subtract and keep sign of bigger number
To get x on its own you ÷ whatever is stuck to it into the other side
Example:
5x-2(3-x) = 2(x+2)
Multiply out brackets
5x -6 +2x = 2x+4
x`s on the left, numbers on the right
5x+2x-2x =4+6
Tidy up x`s and tidy up numbers
5x =10
X = 10
5
X=2
To get x on its own you ÷ whatever is stuck to it into the other side
Example:
3(x-5)-2(1-x) =3-3(4-x)
Multiply out brackets
3x -15-2+2x =3-12+3x
3x+2x-3x=3-12+15+2
2x = 8
X = 8
2
X= 4
x`s on the left, numbers on the right
Tidy up x`s and tidy up numbers
To get x on its own you ÷ whatever is stuck to it into the other side
Example:
-3(x-1)+5=2(x+1)-3(5x-1)+13
Multiply out brackets
-3x+3+5 =2x+2-15x+3+13
-3x-2x+15x = -3-5+2+3+13
10x = 10
X= 10
10
x`s on the left, numbers on the right
Tidy up x`s and tidy up numbers
To get x on its own you ÷ whatever is stuck to it into the other side
X = 1
Example:
11 =7(x+1)-2(3-8x) -3x
11 = 7x+7 -6 +16x -3x
-16x-7x +3x =-11+7+-6
-20x = -10
X = -10
-20
X = + 1 or 0.5
2
Multiply out brackets
x`s on the left, numbers on the right
Tidy up x`s and tidy up numbers
To get x on its own you ÷ whatever is stuck to it into the other side
Evaluating Expressions
Substituting a letter is usually replacing it with a number
If p=2, q=-1, r=3, s=-2, u=4,
Eg. Find the value of
p2+2pr+r2
(2) 2+2(2)(3)+(3)2
Work out the brackets(type into calculator once subbed in)
4 + 12 + 9 = 25
Eg.
u+p= 4+2 = 6 =2
r
3
3
eg.
2(r+2p) = 2(3+2(-1)) = 2(3-2) = 2(1) = 2
pu-3pq
2(4)-3(2)(-1)
8+6
14
14
= 1
7
Addition and subtraction of fractions
When you have 2 fractions to express as 1 fraction you must get the common
denominator
X+2 + x+5
3
4
common denominator =12
4(x+2)+3(x+5)
12
Divide each denominator into 12 and write answer on top
beside whats already on topline
4x+8+3x+15
12
Multiply out brackets and tidy
7x+23
12
Eq.
5x-1 + x – 5
4 3 6
Common denominator =12
3(5x-1)+4(x)-2(5)
12
15x-3+4x-10
12
19x-13
12
12÷4=3, 12÷3=4,12÷6=2
Simultaneous Equations
Are used to find where 2 lines meet and x and y
1. Write both equations with x`s underneath each other and y`s and numbers.
2. Multiply 1 or both equations by a number in order to cancel the x`s or y`s. The
signs must be different too.
3. Add and subtract depending on signs
4. Solve the equation to find x or y
5. When you find 1 value sub back into any of the equations to find the other
value.
Eg.
5x+6y=19
x-2y=-9 X(3)
cancel y`s multiply bottom line by 3
5X+6Y=19
3X-6Y=-27
8X = -8
X= - 1
x-2y=-9
-1-2y=-9
-2y=-9+1
Y`S cancel add or subtract whats left
Replace x with -1 into bottom line
>>> -2y =-8
Y=4
Example:
3x +5y =26
X=3y-10
in wrong place so move 3y to the left
3x+5y=26(X3)
x-3y=-10 (X5)
Cancel y`s by multiplying top line by 3
multiplying bottom line by 5
9x+15y=78
5x-15y=-50
14x =28
X=2
y`s cancel and add and subtract the other terms
X=3y-10
2=3y-10
2+10=3y
12=3y
4=y
Example:
x +y=5
2 3 2
3x-4y=-3
Get common denominator of 6
3x+2y=5(3) = 3x+2y=15
6
3x+2y=15
3x-4y=-3(x-1)
cancel x`s by multiplying bottom line by -1
3x+2y=15
-3x+4y=3
6y=18
Y=3
cancel x`s
3x-4y =-3
3x-4(3) =-3
3x-12=-3
3x=-3+12
3x=9
X=3
sub value for y back in and find x
Changing the Subject of Formula
Putting one letter on one side and everything else on other side and make sure the
letter is on its own.
Example:
H=2k-2
H+2=2k
H+2 = k
2
Example:
p = qr
1
q-r
K=?
move-2 over to get +2
need k on its own so ÷ whatever is with k into other side
r=?
p(q-r)= qr
cross multiply to get on 1 line
pq-pr = qr
r on one side everything else on other side. Move pr
pq =qr+pr
pq = r(q+p)
take out r
pq = r
÷ whatever is with r into other side
q+p
Example
a = b - 3c
1 2 4
b=?
4(a) =2(b) –1(3c)
4
4a = 2b-3c
Get common denominator
4+3c=2b
move 3c over because b plus on right
4+3c = b
2
divide 2 into other side
take only top line
Word Equations
Let x equal to 1 number you don’t know
Let y equal to the other
Use Simultaneous equations to solve
Some hens and a herd of cows are in a field. Between them they have 50 heads and
180 legs. How many cows and hens do you have?
X=hens
X + Y = 50
2X+4Y=180
X + Y = 50(-2)
2X+4Y=180
-2x-2y=-100
2x+4y =180
2y = 80
Y =40
y=cows
Total of cows and hens heads
2 Legs on a hen and 4 legs on a cow
multiply top line by -2 to cancel x`s
cancel x`s and add and subtract what`s left
X + y = 50
X + 40 = 50
X = 10
So 10 hens and 40 cows were in the field
Example
A small bag of cement weighs x kg. A large bag of cement is 4 times as heavy as the
small bag. A medium bag of cement is 5kg lighter than the large bag. Two small bags
plus a heavy bag of cement weigh the same a two medium bags. How much does each
bag weigh?
Let small bag =x
Large bag =4x
Medium bag =4x-5
Small + small+ heavy = medium +medium
X + x + 4x
= 4x-5 + 4x-5
6x
= 8x -10
-8x+6x
= -10
-2x
= -10
X
= 5
small =5kg , large =4(5) =20kg , medium=4(5)-5=15kg
Example
Amy gets €x a prize Brendan gets 3 times as much as Amy and Chloe gets half as much
as Brendan. If the total money received is €550 how much does each person get?
Amy= x
Brendan= 3x
Chloe = 3x
2
X+3x+3x =550
2
use common denominator of 2
2(x)+2(3x)+1(3x)=2(550)
2
2x+6x+3x=1100
11x=1100
X= 1100/11
X=100
Amy=100,
Brendan =300, Chloe=150
Linear Inequalities



Natural numbers =N, (whole numbers start at 1 upwards) Dots on numberline
Integers= Z ,(minus and positive whole numbers)
Dots on numberline
Real Numbers=R ,(All numbers)
Big black line
Eq. 4(x-2)>5(2x-1)-9
4x-8>10x-5-9
-8+5+9>10x-4x
6>6x
6/6>x
1>x
Xϵ R
On numberline
(big black line not including 1 so put a circle)
Eg. 7x-11<2x+9
Put x`s on the right cause positive there and numbers on left
If a term moves direction over inequality it changes signs
tidy up terms
to get x divide whatever is with it into other side
read from x(x less than 1) mouth closed
XϵN
7X-2X<9+11
5X<20
X<20/5
X < 4 (x less than 4 )
Draw numberline
·· ·
Eg. 0 ≤ 9-3x < 9, x ϵR
0 ≤ 9-3X
9-3X <9
3x ≤ 9
9-9 < 3x
X≤ 3
0 < 3x
0< x
Split it down the middle
Factorising
3 different types of factorising
1. HCF (HIGHEST COMMON FACTOR) Take out whats common
2. Difference of 2 squares ax2-by2 everything can be √ always has a minus in the middle
3. Quadratic ax2+bx+c has 3 terms an x2 ,an x and a number
Eg. HCF
9x2-15x
Take out whats common all across 3x
3x(3x-5)
Eg. Difference of 2 squares
36x2-25
(6x-5)(6x+5)
√ each number and split them into the brackets
different signs in each bracket
27x2-3y2
3(9x2-y2)
3(3x-y)(3x+y)
Do HCF first
Then difference of 2 squares
put it all together
Eg. Quadratic
3x2+10x+8
Get Factors of 3 ---3X1, Get Factors of 8---4X2 or 8X1
Ans: (3x+4)(x+2)
+ for last sign and 1st sign means you have 2 ++ and you add
Check everything is in the correct place with signs by
Multiplying terms together.
+6x
+4x
10x
value in the middle with sign
Or Array Method (boxes)
signs + +
3x2+10x+8
Multiply 3X8=24 get factors of 24 that add to give you 10
3x2+6x+4x+8
Get factors 6X4 or 8X3 or 2X12 or 24X1
Write everything in a box
3x
3x2
4x
3x
8
X
2
What multiplied by 3x gives you 3x2
3x2
6x
What multiplied by 3x gives you 6x
4x
4
Take out 3x on top row using HCF
6x
8
What multiplied by x gives you 4x
Take the top of table and side for factors
(3x+4)(x+2)
Same answer
Simply by factorising
x2+x-6
X2+3x-10
x2+x-6 = (x+3)(x-2)
x2+3x-10 = (x+5)(x-2)
(x+3)(x-2)
(x+5)(x-2)
cancel x-2 on top and bottom
x+3
X+5