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. -8*-2=+16
• Minus*plus= minus eg.-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-6y2Add and subtract like terms

6x2+6xy

Multiplying out Brackets

eg. (x+5)(x-3) Can use boxes (array method)

X2 / -3x
5x / -15

X-3

X

+5

X2-3x+5x-15Ans: X2+2x -15

1st term 2nd term

0r use (x+5)(x-3)2nd bracket

1stterm(2nd bracket) +2nd term (2nd bracket)

x (x-3) + 5 (x-3)

X2- 3x + 5x-15

X2+2x -15 (same answer as above)

Multiplying out Brackets

Eg. (5a-2)2

(5a-2)(5a-2)

25a2 / -10a
-10a / +4

5x-2

5a

-2

25a2-10a-10a+4

25a2-20a+4

1st term 2nd term

0r use (5a-2)(5a-2)2nd bracket

1stterm(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)Multiply everything inside by what`s inside brackets including signs

-4x + 4+ 2x- 16Tidy 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

1m2-14m-1

Solving Equations

Follow the steps to solve

1. Multiply out brackets
2. Put x`s on the left of the equals & numbers on the right(anything moves sides it changes sign)
3. Tidy up x`s and tidy up numbers
4. To get x on its own you ÷ whatever is stuck to it into the other side.

Example:

5x - 3= 17x`s on left numbers on right move -3 and becomes+3

5x = 17+3tidy up numbers-- 2 same signs add and keep sign

5x = 20to get x ÷ 5 into 20

x = 20

5To get x on its own you ÷ whatever is stuck to it into the other side

x = 4

Example:

4x-2 = 5x -5x`s on left changes to -5x numbers on right move -2 and becomes+2

-5x+4x = -5+22 different signs subtract and keep sign of bigger number

-1x = -3

X = -3Toget x on its own you ÷ whatever is stuck to it into the other side

-1

X = 3

Example:

5x-2(3-x) = 2(x+2)Multiply out brackets

5x -6 +2x = 2x+4x`s on the left, numbers on the right

5x+2x-2x =4+6Tidy up x`s and tidy up numbers

5x =10

X =10To get x on its own you ÷ whatever is stuck to it into the other side

5

X=2

Example:

3(x-5)-2(1-x) =3-3(4-x)Multiply out brackets

3x -15-2+2x =3-12+3xx`s on the left, numbers on the right

3x+2x-3x=3-12+15+2Tidy up x`s and tidy up numbers

2x = 8

X = 8To get x on its own you ÷ whatever is stuck to it into the other side

2

X= 4

Example:

-3(x-1)+5=2(x+1)-3(5x-1)+13 Multiply out brackets

-3x+3+5 =2x+2-15x+3+13x`s on the left, numbers on the right

-3x-2x+15x = -3-5+2+3+13Tidy up x`s and tidy up numbers

10x = 10

X= 10To get x on its own you ÷ whatever is stuck to it into the other side

10

X = 1

Example:

11 =7(x+1)-2(3-8x) -3xMultiply out brackets

11 = 7x+7 -6 +16x -3xx`s on the left, numbers on the right

-16x-7x +3x =-11+7+-6Tidy up x`s and tidy up numbers

-20x = -10

X = -10To get x on its own you ÷ whatever is stuck to it into the other side

-20

X = + 1 or 0.5

2

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)2Work out the brackets(type into calculator once subbed in)

4 + 12 + 9 = 25

Eg.

u+p= 4+2 = 6 =2

r3 3

eg.

2(r+2p) = 2(3+2(-1)) = 2(3-2) = 2(1) = 2 = 1

pu-3pq 2(4)-3(2)(-1) 8+6 14 14 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+5common denominator =12

3 4

4(x+2)+3(x+5)Divide each denominator into 12 and write answer on top

12beside whats already on topline

4x+8+3x+15Multiply out brackets and tidy

12

7x+23

12

Eq.

5x-1 + x – 5Common denominator =12

4 3 6

3(5x-1)+4(x)-2(5)12÷4=3, 12÷3=4,12÷6=2

12

15x-3+4x-10

12

19x-13

12

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=-27Y`S cancel add or subtract whats left

8X = -8

X= - 1Replace x with -1 into bottom line

x-2y=-9

-1-2y=-9

-2y=-9+1 > -2y =-8 Y=4

Example:

3x +5y =26

X=3y-10in wrong place so move 3y to the left

3x+5y=26(X3)Cancel y`s by multiplying top line by 3

x-3y=-10 (X5)multiplying bottom line by 5

9x+15y=78y`s cancel and add and subtract the other terms

5x-15y=-50

14x =28

X=2

X=3y-10

2=3y-10

2+10=3y

12=3y

4=y

Example:

x +y = 5

2 3 2

3x-4y=-3

3x+2y=15cancel x`s by multiplying bottom line by -1

3x-4y=-3(x-1)

3x+2y=15cancel x`s

-3x+4y=3

6y=18

Y=3

3x-4y=-3sub value for y back in and find x

3x-4(3)=-3

3x-12=-3

3x=-3+12

3x=9

X=3

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-2K=?

H+2=2kmove-2 over to get +2

H+2 = kneed k on its own so ÷ whatever is with k into other side

2

Example:

p = qrr=?

1 q-r

p(q-r)= qrcross multiply to get on 1 line

pq-pr = qrr 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- 3cb=?

1 2 4

4(a) =2(b) –1(3c)Get common denominator

4

4a = 2b-3ctake only top line

4+3c=2bmove 3c over because b plus on right

4+3c =bdivide 2 into other side

2

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=hensy=cows

X + Y = 50Total of cows and hens heads

2X+4Y=1802 Legs on a hen and 4 legs on a cow

X + Y = 50(-2)multiply top line by -2 to cancel x`s

2X+4Y=180

-2x-2y=-100

2x+4y =180cancel x`s and add and subtract what`s left

2y = 80

Y =40

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 = 5small =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 use common denominator of 2

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)-9Xϵ R

4x-8>10x-5-9Put x`s on the right cause positive there and numbers on left

-8+5+9>10x-4xIf a term moves direction over inequality it changes signs

6>6xtidy up terms

6/6>xto get x divide whatever is with it into other side

1>xread from x(x less than 1) mouth closed

On numberline(big black line not including 1 so put a circle)

Eg. 7x-11<2x+9XϵN

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 ≤ 9-3X9-3X <9Split it down the middle

3x ≤ 99-9 < 3x

X ≤ 30 < 3x

0 < x

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 has3 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√ each number and split them into the brackets

(6x-5)(6x+5)different signs in each bracket

27x2-3y2Do HCF first

3(9x2-y2)Then difference of 2 squares

3(3x-y)(3x+y)put it all together

3x2+10x+8Get 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

+6xMultiplying terms together.

+4x

10xvalue in the middle with sign

Or Array Method(boxes)signs+ +

3x2+10x+8Multiply 3X8=24 get factors of 24 that add to give you 10

3x2+6x+4x+8Get factors 6X4 or 8X3 or 2X12 or 24X1

Write everything in a box

3x2 / 6x
4x / 8

3xTake out 3x on top row using HCF

X2What multiplied by 3x gives you 3x2

3x2 / 6x
4x / 8

3xWhat multiplied by 3x gives you 6x

4

What multiplied by x gives you 4x

Take the top of table and side for factors

Simply by factorising

x2+x-6x2+x-6 = (x+3)(x-2)

X2+3x-10x2+3x-10 =(x+5)(x-2)

(x+3)(x-2)cancel x-2 on top and bottom

(x+5)(x-2)

x+3

X+5