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/4
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 a
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
a2 and 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 ⅓ 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
