A variable is a symbol that is used to represent an unknown quantity. For example, x, y, z. If x and y are variables, then the product xy is also a variable.
A term can be:
a single number. For example, 2, 5.
a variable, or a product of variables (which may be raised to powers).
For example, z, b3, [pic].
a product of a number and one or more variables (which may be raised to powers).
For example, 3x, – 4yz, 7a2 b3 .
In a term that is the product of a number and one or more variables, the number is called the numerical coefficient or simply the coefficient. For example, in the term –2b3, the coefficient is –2 and the variable part is b3.
Like and unlike algebraic terms
Like algebraic terms are defined as those terms which are represented by the same algebraic symbol, regardless of the sign or the magnitude of their coefficients. Thus 5x, –3x, [pic] and [pic] are like algebraic terms since they are all represented by the same symbol x. Similarly 3a2 , –2a2, 0.4a2 and [pic]a2 are like terms.
Unlike algebraic terms are terms that are represented by different algebraic symbols. Thus 7b, 3b2 and –2b3 are all unlike terms even though they are powers of the same variable, b. Similarly, [pic], [pic]and [pic], read as “y one”, “y two” and “y three” respectively, are unlike algebraic terms and are interpreted as the “first y”, the “second y” and the “third y” .
When we combine numbers and variables with the ordinary operations of arithmetic (in some meaningful way), the result is called an algebraic expression. Addition/subtraction signs separate algebraic expressions into terms. For example,
(1) 2 + 3x – 4y + 5z, (2) 7a2 b3 + 5, (3) (x – y)(y – z)(z – x), (4) [pic]. The expressions above have no specific value unless we assign values to the variables a, b, x, y, and z. The values of these expressions may vary depending on the values assigned to these symbols. Ex.Find the value of each of the expressions above given that
a = –1, b = 2, x = 0, y = 3, z = –2.
It is important to be able to distinguish between a term of an expression and a factor of a term. Terms are separated by +/– symbols. Factors are numbers and/or variables that are multiplied together. For example, in the expression x + 3xy, x and 3xy are terms, because they are separated by a + symbol. In the term 3xy, x and y are factors, because x, y, and 3 are multiplied together.
Addition & Subtraction
Only like algebraic terms can be added or subtracted.
Ex.Simplify the following expressions
(1)7x – 8x + 3x(2)[pic](3)[pic](4)[pic](5)3x – 1(6) [pic]
Recall that multiplication is commutative and associative. This is true for numbers as well as variables. Thus ab = ba, [pic].
Note. We do not write “x3”. Instead, we write “3x”. The coefficient is always put first. Ex.Simplify the following:
When dividing we can cancel factors that are common to both the numerator and the denominator. Ex.Simplify the following:
Multiplication distributes over addition and subtraction.
Using brackets we sometimes need to group unlike algebraic terms together. This is where the distributive law is useful, for it can be used both to remove and insert brackets in algebraic expressions.
Ex.Remove the brackets and simplify the following:
(1)3(4x + 3y) + 4(3x + 2y)(2)8(5x + 2y) – 7(3x – 7y)
(3)9(5x – 2y +3z) + (3x + 2y –5z )(–6)(4)[pic]
An equation is a statement that signifies an equality relationship between two quantities or expressions. Such a relationship is denoted by the equal sign “=”. Examples2x + 3 = 4 – 8x,[pic],[pic]
Note that an...