Basic Linear Algebra

1. 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 INTO Foundation L3 MH

2. Starter - add adjacent expressions to make a new expression in the cell above 25x - 5 12x -2 13x - 3 6x - 2 5x - 1 7x - 1 2x +1 4x +1 3x -2 2x -3

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

4. … 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?

5. Basic Algebra In Algebra we • Use symbols to represent unknown numbers • We solve equations to find the values of these unknown quantities An algebraicEXPRESSIONconsists of one or more TERMSseparated by +/- operators TERMS EXPRESSION INTO Foundation L3 MH

6. Collecting Terms • We should always reduce algebraic expressions to their simplest form as an answer • We can only collect like Terms Simplifies to INTO Foundation L3 MH

7. Brackets • 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 INTO Foundation L3 MH

8. Removing Brackets 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 INTO Foundation L3 MH

9. Removing Brackets Simplify expand brackets collect terms INTO Foundation L3 MH

10. Multiplication of Terms This is where our knowledge of the laws of indices comes in handy Division similar Example INTO Foundation L3 MH

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

12. Simplifying Algebraic Fractions • We do this in exactly the same way as with numerical fractions Example INTO Foundation L3 MH

13. Fractions Example Example INTO Foundation L3 MH

14. Dividing Fractions • Simply Invert the divisor and multiply INTO Foundation L3 MH

15. 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 INTO Foundation L3 MH

16. 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!! INTO Foundation L3 MH

17. 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 INTO Foundation L3 MH