Download 5.1-2 An Introduction to Algebra

Introduction to Algebra
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, a variable of x and
x has a 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 coefficients
and x and y are the 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 = 5y
5w + w = 6w
5.1x - 3.4x = 1.7x
69a - 47a = 22a
•
•
•
•
2x - 6x + 5 = -4x + 5
-4y + 8 – y = -5y + 8
-6x - 3 - 5x = -11x - 3
2x + 3y - x + 9y = x + 12y
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.
7q  6  4  7 q  42  4
 7 q  38
Simplify the expression:
2  4 x  3  2 = 8x – 6 + 2
= 8x – 4
6  4 y  2   7 = 24y + 12 – 7
= 24y + 5
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.
2  (5  8t )  2  5  8t
 3 8t
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 side and the right side.
In the equation
2x + 3 = 13,
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 • 8 + n = 34
is 34.
• 5 less than 3 times a
number is 45.
• The sum of 15, 4 and
a third number is
zero.
• 3x – 5 = 45
• 15 + 4 + t = 0