Saturday, 1 February 2020

S1T1 Algebra Notation, Expressions and Formulae

Algebra: The use of letters (a, b, x, y, …) to represent numbers (or quantities)

               The letters are used to denote unknown numbers or variables.


Variables and Algebraic Expressions


Variables ~ letters to represent numbers that we don't know

Algebraic Expressions => numbers and letters that are connected by operations ( x / + -)


Example of Algebraic Expressions

Add y and 4  = y + 4


5 group of a  = 5 x a =  5a


Product of 5, a and b = 5 x a x b = 5ab


Divide y by 4 = y/


Example

Tina has $x at first. She spent $3 on a pen. 

a.   How much money is she left with?

x - 3


b.    Her sister gave her $5. How much money does she have?

x + 5


Equivalent Algebraic Expressions

Properties

   a + b = b + a

           a x b = b x a


Algebraic Expression can be written in different forms.


Example

x + 2 = 2 + x

1/2 ( p + q) = p + q

                                 2

 

Use of Bracket and Order of Operations

Property:


a ( b + c) = ab + ac


Example

 

3 ( a + b ) = 3 x ( a + b ) = ( a + b ) x 3


The preferred written form is 3(x + y) 


Similarly,

½ ( u + v ) = (u + v) = ½ u + ½ v

       2

Example 

(i) Expand        4 (a + b) 

                     = 4 x a + 4 x b 

                     = 4a + 4b

            

(ii) Expand        4 + a

                            5          

                      =   1 x (4 + a) 

                           5                                 

                      = 4 + 1

                         5    5

                      = 4 + a

                         5    5


Example

Mary has $p in her wallet and $q in her pocket.

She spent half of the total amount of money and gave 20% of the total amount to her sister.

Write an algebraic expression for

(a) the amount of money that she spent

      Total amount = p + q

       Spent = 1/2 (p + q)


(b)  the amount to her sister 

             = 20% of total amount

             = 20/100 x (p + q)

     = 1/5 (p + q)


Square and Cube

Area of square of side p = side x side

                       = p x p = p2 , (not written as pp)

Similarly,

Volume of square = p x p x p = p3 


Example 

Write the equivalent Algebraic Expression of 

      2 x p x p x q

      = 2 p2 q


Example

A square has a length of p. 

Write an algebraic expression for 

(a) What is the total area of 6 such square?

     Area of square = p x p = p2    (Step1 : Write the formula)

              6 square = 6 x p2           (Step 2: other operations? => x 6)

             = 6p2 

(b)  What is the volume of the 6 squares?

       Volume of square = p x p x p = p3     (Step 1 : Write the Formula)

Volume of 6 squares = 6 x p3 = 6 p3 (Step 2 : Operation operations?)


Evaluating Algebraic Expressions and Formulas

Example

Simplify 5y – 2(y – 2)

            = 5y – 2y - 2 x -2 (Step1 : Open Bracket)

            = 3y + 4         (Step2: Do x / + - )


Example

Simplify    7y – 2(3y – 5) 

              = 7y – 6y + 10 

              = y + 10


Example

Find the value of 10 – 3k for k = 4

            10 – 3(4) = 10 – 12 = -2 (Step : Substitute k = 4 into k)


Example

The cost of parking, $p, at a car park is related to the parking duration, t hours, by the formula 

p = 2 + 2.5t

Find the value of p for each of the following values of t.

a.     t = 1

p = 2 + 2.5(1) = 4.5

b.     t= 3

p = 2 + 2.5(3) = 2 + 7.5 = 9.5


Example

Find the value of 12x + 2xy when x = 2 and y = 5

             12x2 + (2 x 2 x 5)

            = 24 + 20

            = 44


In Summary

1.  ab = a x b      


2.  a/b => a ÷ b => a x 1

                                    b

3.  a2= a x a,           a3= a x a x a


4.  a2b = a x a x b,            ab2 = a x b x b

            

5.  3p = 3 x p = p + p + p 


6.  3(p + q) = 3 x (p+ q) = 3p + 3q


7.  2(3 + y) =  2(3 + y) ÷ 5 =  2(3 + y)

     5                                           5


8. a2 + a= a(a + 1)        a2   = a x a, 


Note:                                     

a2 = a x a 

2a = 2 x a and 2a 

        aand 2a are 2 different algebraic terms.


Properties:

   a + b = b + a

           a x b = b x a


Practice

1.  Product of 4 and c and f


2.  Subtract  p from r and multiply the result by 3.


3.  Divide p by q


4.  Divide the square of x from the difference of y and x.


5.  Simplify the following algebras

     (a)   x + 3x + 4x          

     (b)   x + y + 3x + 2y

     (c)   -2z - 2z         

     (d)   2x2 + 2x + x - x2     

     (e)   7x2 + 3x - 8x - 4x2

     (f)    -x2 - 3x - x + x2


6. A box contains p pens and q pencils. How many pen and pencils are there in 6 boxes?


7. Annie has $p. She earns $q. She then spends of the total amount of money and save 20% of the total amount.

Write an algebraic expression for

a. the amount of money that she spends

b. the amount of money that she saves.


8.   Kenny's daily pay is $p, and is related to his  overtime hours, t by the formula

p = 100 + 12t

      Find the value of p for t when (a) t = 0, (b) t = 4.5


9. If a = 2, b = 3 and c = -2, evaluate the following:

  (a)  3a + b     (b)  5c - 2b     (c)  ab - 5    (d) ab2

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