Five years ago the ratio of Samwel's age to that of Juma was 5:4. Ten years from now the ratio of their ages will be 10:9. Find the present age of Samwel and Juma.
ALGEBRA
FORM TWO STUDENTS
Introduction.
This is a branch of mathematics in which arithmetic operations and formal manipulations are applied to abstract symbols rather than specific numbers. This course introduces you into arithmetic operations and formal manipulations of expressions and equations involving more than two variables.
Algebraic Expressions.
An algebraic expression is an expression involving variables as well as numbers. An algebraic expression involves sum, difference, product or quotient of numbers and variables. Numeral and variables in an algebraic expression separated by plus or minus signs are called terms. Numerals and letters that are multiplied to give the terms are called factors of the term and the numerical factor is called co-efficient of the term.
Example 1.
7x + 6, this is an expression with two terms i.e 7x and 6.
7 and x are factors of the term 7x
7 is the co-efficient of x.
Example 2.
Study the following expression.
A variable is a symbol for a number we don’t know yet. It is normally a letter, for example x or y.
A constant is number on its own.
Binary Operations on expressions.
Binary operation is the operation of two mathematical objects involving addition, subtraction, multiplication and division and put together. The following are examples of binary operations: 3a – b, 7a + 3b, 8 + 6x, 9 + 2, 3y – 2x, etc.
When one or more terms have exactly the same variable with the same power, they are called like terms. When the variables are different, they are called unlike terms.
Example, x, 2x and -3x are al terms in x and therefore like terms whereas 3, 3x2, 4xy are unlike terms since each term contains a different power of x.
Note: The binary operation may be denoted by symbols such as ×, ∆, * and so on depending the instruction given to the operation.
Operations.
Example 1. Evaluate 5 × 123
Solution:
5 × 123 = 5 × (100 + 20 + 3)
= (5 × 100) + (5 × 20) + (5 × 3)
= 500 + 100 + 15
= 615
Example 2. Evaluate (7 × 98)
Solution:
7 × 98 = 7 × (100 – 2)
= (7 × 100) – (7 × 2)
= 700 – 14
= 686
Example 3. Evaluate (3 × 46) + (3 × 54)
Solution:
(3 × 46) + (3 × 54) = 3 × (46 + 54)
= 3 × 100
= 300
Example 4. Evaluate (8 × 89) – (8 × 79)
Solution:
(8 × 89) – (8 × 79) = 8 × (89 – 79)
= 8 × 10
= 80
Example 5. Simplify the following algebraic expressions.
(a) 2x + 4y – 3x + 6 + 6y
(b) 3a – 6b + 8a – 2b + a
Solution:
(a) In order to perform 2x + 4y – 3x + 6 + 6y, identify the terms and collect like terms together as follows.
2x + 4y – 3x + 6 + 6y
= 2x – 3x + 4y + 6y + 6
= - x + 10y + 6
Therefore, 2x + 4y – 3x + 6 + 6y = -x + 10y + 6
(b) 3a – 6b + 8a – 2b + a
Grouping like terms together
= (3a + 8a + a) + (-6b – 2b)
= 12a – 8b
Multiplication of algebraic expressions.
Recall the rules for multiplying integers by positive and negative numbers. These rules are:
a. (+)× (+) = (+)
b. (+)× (-) = (-)
c. (-) × (+) = (-)
d. (-) × (-) = (+)
These rules for multiplying directed numbers also apply to algebra as well.