7.1variable notation. in arithmetic, we perform mathematical operations with specific numbers. in...
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7.1Variable Notation
In arithmetic, we perform mathematical operations with
specific numbers. In algebra, we perform these same basic
operations with numbers and variables- letters that stand for
unknown quantities.
Algebra is considered to be a generalization of arithmetic. In
order to do algebra it is important to know the vocabulary and
notation (symbols) associated with it.
An algebraic expression consists of constants , variables , and
operations along with grouping symbols .
The numerical coefficient of a variable is the number that is multiplied by the variable. For example, the expression 2x + 5 has constants of 2 and 5,
variable of x and x has coefficient of 2.
The terms of an algebraic expression are the quantities
that are added (or subtracted).
When a term is the product of a number and letters or letters alone,
no symbol for multiplication is normally shown. For example 2x
means 2 times some number x and abc means some number a times
some number b times some number c.
Constants are numbers which do not change in value. Variables are unknown
quantities and are represented by letters.
In the expression 2x +3y -5,the 2, 3, and 5 are constants and x and y are variables.
To evaluate an algebra expression, substitute
numbers for the variables and simplify using the order of
operations. It is a good idea to replace the variables with their
values in parentheses.
For example to evaluate 2x - y when x = 5 and y = -3, replace the variables with their values
in parentheses 2(5) - (-3)
then simplify.
10 + 3 = 13
Terms are always separated by a plus (or minus) sign not inside parentheses. The expression 2x - 3y has two terms, 2x and -3y. 2 and -3 are constants, x
and y are variables with 2 being the coefficient of x and -3 the
coefficient of y.The expression 2x +3y -5 has 3
terms.
LIKE TERMS are terms whose variable factors are the
same. Like terms can be added or subtracted by adding (subtracting) the coefficients. This is sometimes referred to
as combining like terms.
Example: Simplify each expression by combining like
terms.
• 7y - 2y
• 5w + w
• 5.1x - 3.4x
• 69a - 47a - 51a
• 2x - 6x + 5
• -4y + 8 - y
• -6x - 3 - 5x -4
• 2x + 3y - x +9y
If an algebraic expression that appears in parentheses cannot be simplified, then multiply each term inside the parentheses by
the factor preceding the parentheses, then combine like
terms.
Example: Simplify the expression by combining like
terms.
467 q4427 q
387 q
Simplify the expression:
2 4 3 2x
6 4 2 7y
If an expression inside parentheses is preceded by a “+” sign, then
remove the parentheses by simply dropping them. For example:
3x + (4y + z) = 3x + 4y + z
If an expression in parentheses is preceded by a “-” sign then it is removed by
changing the sign of each term inside the parentheses
and dropping the parentheses.
3x – (4y – z) = 3x – 4y + z
Example: Simplify the expression by combining like terms.
)85(2 tt852
t83
An equation is a statement that 2 expressions are equal. The symbol “=“ is read “is equal to” and divides the equation into 2 parts, the left member and the right member. In
the equation 2x + 3 = 13,
2x + 3 is the left member and 13 is the right member.
The solution to an equation in one variable is the number that can be substituted in place of the variable and makes the equation true.
For example 5 is a solution to the equation 2x + 3 = 13
because 2(5) + 3 = 13 is true.
To solve an equation means to find all solutions or roots for the
equation.
Solve each equation:
• z = 4 + 9
• p = 3(9) – 5
• b = 5(3) – 4(8) + 7
To write a verbal statement into a symbolic statement:
• Assign a letter to represent the missing number.
• Identify key words or phrases that imply or suggest specific mathematical operations.
• Translate words into symbols.
Write the statements into symbols:
• 8 more than a number is 34.
• 5 less than 3 times a number is 45.
• The sum of 15, 4 and a third number is zero.
• 8 + n = 34
• 3x – 5 = 45
• 15 + 4 + t = 0