Basic Linear Algebra Objectives Manipulating algebraic Terms and Expressions Collecting like Terms Removing brackets Factorising Multiplication of Algebraic Terms Simplifying Algebraic Fractions Dividing Fractions Manipulating algebraic equations Solving a single linear equations 24 October, 2019 INTO Foundation L3 MH

Starter - add adjacent expressions to make a new expression in the cell above

17t +7 9t +8 7t -1 3t-8 5t +1 4t +7 2t +3 3t -2 t +9 2t -17

… Starter eggs …? … I went to the market with a basket of eggs… I sold half my eggs plus half an egg to Alan I sold half of my remaining eggs plus half an egg to Barry I sold half of my remaining eggs plus half an egg to Charlie I sold half of my remaining eggs plus half an egg to Denise I sold half of my remaining eggs plus half an egg to Ellen I was left with just one egg. Are the above transactions possible without breaking eggs? How many eggs did I start with?

Basic Algebra In Algebra we Use symbols to represent unknown numbers We solve equations to find the values of these unknown quantities An algebraic EXPRESSION consists of one or more TERMS separated by +/- operators TERMS EXPRESSION 24 October, 2019 INTO Foundation L3 MH

Collecting Terms We should always reduce algebraic expressions to their simplest form as an answer We can only collect like Terms Simplifies to 24 October, 2019 INTO Foundation L3 MH

Brackets Example 4t+ 16ty can be written as 4t(1+4y) We can use brackets to factorise an expression Take out a common factor Example 4t+ 16ty can be written as 4t(1+4y) Brackets can also be thought of as simplifying an expression Brackets can also be used to explicitly define the order of operations Find the value of 2x+4xy and 2(x+4xy); when x=2,y=3 24 October, 2019 INTO Foundation L3 MH

Removing Brackets a(b+c) ≡ ab + ac This is the opposite to factorisation The factor outside the bracket multiplies each Term inside. a(b+c) ≡ ab + ac -a(b-c) ≡ -ab +ac (Same as arithmetic ) We do not explicitly write the multiplication term in algebra because it looks too much like an “ “ Use the curly “ “ , although on the slides I sometimes use the “x” for convenience 24 October, 2019 INTO Foundation L3 MH

Removing Brackets Simplify expand brackets collect terms 24 October, 2019 INTO Foundation L3 MH

Multiplication of Terms This is where our knowledge of the laws of indices comes in handy Division similar Example 24 October, 2019 INTO Foundation L3 MH

Division of Terms Example : Simplify We normally write an expression like this as a fraction Then we cancel variables 24 October, 2019 INTO Foundation L3 MH

Simplifying Algebraic Fractions We do this in exactly the same way as with numerical fractions Example 24 October, 2019 INTO Foundation L3 MH

Fractions Example Example 24 October, 2019 INTO Foundation L3 MH

Dividing Fractions Simply Invert the divisor and multiply 24 October, 2019 INTO Foundation L3 MH

Summary so far Looked at handling simple algebraic expressions, terms and fractions Do exercise ->Section 1 Basic Algebraic Manipulation and solving equations Q1-Q6 Next solving simple Algebraic equations 24 October, 2019 INTO Foundation L3 MH

Single linear equations -This is generally very straight forward -Essentially all you need to do is rearrange the equation in terms of the unknown variable -As long as you do the same thing to both sides an equation will still be valid Lets make up some examples!! 24 October, 2019 INTO Foundation L3 MH

2 quick Examples 3(x+2)=24 4(x+2)=9(x+2) Now Do: Section 1 Basic Algebraic Manipulation and solving equations Q7- onwards 24 October, 2019 INTO Foundation L3 MH