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ALGEBRA

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ALGEBRA(Topics) Algebra Basics- Addition, Subtraction and Substitution. Algebra Basics 2- Multiplication, Expansion of Brackets and Division Indices Equations and Inequalities Simultaneous Equations Simultaneous Equations in solving equations

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ALGEBRA(Topics) Functions and Rearranging. Quadratic equations

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ALGEBRA(Key Terms) Substitution- Evaluate- Variable- Constant- Simplify- Coefficient- => means ‘this gives’ nb: means ‘note well’

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ALGEBRA(Basics) The four basic operations for addition, subtraction, multiplication and division are carried out on variables much like on numbers. One practical way to look at it is if I have a variable ‘a’ and I am adding ‘2a’ to it. Imagine a=‘apples'. Ask yourself if I have one apple and someone gives me two how many do I have. Questions in the exam normally come in the form:- a+2a This gives 3a.

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ALGEBRA(Basics) The same is done with subtraction If I have 5 mangoes and my grandmother takes 3 mangoes how many do I have. Let ‘m’ be equal to mangoes(step 1) Represent it in a mathematical statement(step 2) 5m-3m= Then Evaluate the statement 5m-3m=2m.(step3)

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ALGEBRA(Basics) Sometimes Variables have negative coefficients. For example, You were playing cricket and you lost the ball by hitting a massive ‘6’ you now owe one of your friends a ball. Hence you have -balls(or negative 1 balls). The next day your mother buys a three pack of balls but you still owe your friend, this situation can be represented in a mathematical statement as -b+3b This means you actually own only 2 of those balls, sine you owe you friend. Therefore –b+3b=2b.

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ALGEBRA(Basics) This statement can be rewritten as 3b+ - b which is equal to 3b-b which is equal to 2b.

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ALGEBRA(Basics) When writing mathematical statements or expressions variables of different types cannot be combined or simplified further. If you have three balls and two apples you can represent it as 3b+2a where b=‘balls’ and a=‘apples’

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ALGEBRA(Basics) You may be given an expression like eg.1 -b+3a+2b-4a And asked to simplify the expression. Bring similar(or like) variables together, that is ‘a’s with ‘a’s and ‘b’s with ‘b’s. Take whatever sign is in front of them and rewrite the statement. Do one variable first then the next.(step 1) -b+2b+3a-4a

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ALGEBRA(Basics) -b+2b+3a-4a Again negative one ‘b’ plus two b, is one b as if subtracting. Three ‘a’ minus four ‘a’ give negative one ‘a’. So we can simplify the expression as b+ - a which is equal to b-a

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ALGEBRA(Basics) eg.2 3p-4q+6p+2q Collect like terms with the same sign and rewrite =3p+6p-4q+2q simplify(remember-4p+2q =2q+ - 4q =2q-4q =-2q) =9p-2q

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ALGEBRA(Basics) eg.3 5c-2d+7d-3c Collect terms and rewrite with the same signs in front of them =5c-3c-2d+7d Simplify(remember -2d+7d =7d+ - 2d =7d-2d =5d) =2c+5d

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ALGEBRA(Basics) Sometimes we are given values for our variables we can Substitute the to get actual values for our expressions. eg.1 Given x=3, y=2 Evalutate x+y Replace x with the value given for x and y with the value given for y and rewrite(Step 1) NB: it is best to use brackets.

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ALGEBRA(Basics) Evaluate (Step 2) x+y = (3)+(2) =5 We will show why brackets are important further down in this Topic. eg.2 Given a=2 and b=1 Evaluate 2a+2b Replace(Substitute) ‘a’ with 2 and ‘b’ with 1 in the expression.(Step 1) Expand Brackets and Evaluate(Step 2) =>2(2)+2(1) =4+2 =6

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ALGEBRA(Basics) eg.3 Given c=-1 and d=2 Evaluate c+3d =(-1)+3(2) => -1+6 (NB:6+-1 = 6-1) =5 eg.4 Given e=2 and f=-3 evaluate e-2f Rewrite expression as it is just substituting the variables =>(2)-2(-3) =>2-(-6) =8 Expand Brackets, then evaluate We get 8 because if I take away some thing negative like debt it is as if I have added to what you have. Imagine you owe your father $4 that is -$4 you have. If your mother convinces him to clear the debt she has taken away the -$4. -$4-(-$4)=0 Like adding -$4+$4=0

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ALGEBRA(Basics) Algebra Basics Ends Here go to Algebra Exercise section and do Basic revision exercise.

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ALGEBRA(Basics 2) There is a general sequence when working with Algebraic expressions involving which operations to conduct first. B.O.D.M.A.S B.O.D.M.A.S – Brackets (operated on before),Division(then),Multiplication(then), Addition(then),Subtraction.

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ALGEBRA(Basics 2) eg.1 Simplify 3y-2(y+4)= According to BODMAS let us expand brackets first. That is take the coefficient in front the bracket and multiply it by everything inside the bracket. The coefficient is -2 so by expanding the bracket we get 3y+ - 2(y)+ - 2(4) {nb: rewrite sign the bracket} =>3y+ - 2y+ - 8 =>3y-2y-8 =>y-8

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ALGEBRA(Basics 2) eg.2 Simplify 3x-2(x+5) Remember B.O.D.M.A.S! =>3x+ - 2(x)+ - 2(5) {nb: rewrite sign the bracket} =>3x-2x-10 =>x-10

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ALGEBRA(Basics 2) eg.3 Simplify 5(a-b)+2b =>5(a)-5(b)+2b {nb: rewrite sign the bracket} =>5a-5b+2b =>5a-3b

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ALGEBRA(Basics 2) eg.4 Simplify 3(a-b)-a =>3(a)-3(b)-a =>3a-3b-a By collecting terms =>3a-a-3b =>2a-3b

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Algebra(Indices) Key Terms Index- An index is the power to which a number is ‘expressed’ that the number of times it is multiplied by itself. Eg. x

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