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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