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Algebra Basics
Variables and Expressions
Variable – a symbol used to represent a quantity
that can change.
Coefficient – the number that is multiplied by the
variable in an algebraic expression.
Numerical expression – an expression that contains
only numbers and operations.
Algebraic expression – an expression that contains
numbers, operations and at least one variable.
Constant – a value that does not change.
Evaluate – To find the value of a numerical or
algebraic expression.
Simplify – perform all possible operations including
combining like terms.
Order of operations – a rule for evaluating
expressions from left to right.
Order of Operations
When a numerical expression involves 2 or
more operations, there is a specific order in
which these operations must be performed.
In earlier grades you learned to solve
numerical expressions using “order of
operations” or as most of us say PEMDAS
which stands for
Parenthesis, Exponents,
(Multiplication/Division) then (Add/Subtract)
PEMDAS
Please Excuse (My Dear) (Aunt Sally)
Please Excuse (My Dear)(Aunt Sally)
The reason (multiplication & division) and (add &
subtract) are grouped is when those operations
are next to each other you do the math from left
to right. You do not necessarily do addition first
if it is written next to subtraction.
equation – is a mathematical statement that two
expressions are equal (=).
formula – is an equation that shows a
mathematical relationship between 2 or more
quantities.
reflexive property of equality – for all real
numbers a, a = a.
symmetric property of equality – for all real
numbers a and b, if a = b, then b = a.
transitive property of equality – for all real
numbers a, b, and c, if a = b and b = c, then a = c
Numerical expression
8–6+2
Subtraction is first
Remember when those operations, (multiplication &
division) and (add & subtract) are grouped next to
each other you do the math from left to right.
When there are 2 or more operations, and we
use grouping symbols such as parenthesis or
brackets, you perform the inner most grouping
symbol first.
2 + 3[5 +(4-1)²]
2 +3[5 + (3)²]
2 + 3[5 + 9]
2 + 3[14]
2 + 42
44
Summary of Basic Steps
• Copy the problem.
• Simplify any grouping symbols (such as
parenthesis) first, starting with the inner most
group.
• Simplify any powers (exponents).
• Perform the multiplication & division in order
from left to right.
• Do the addition & subtraction last, from left to
right.
• Remember, if operations are written next to each
other, work from left to right.
It is very important to understand that it does
make a difference if the order is not
performed correctly!!!!!!!!!
70 – 2(5 + 3)
70 – 2(8)
68(8)
544 incorrect
(subtraction was done before multiplication)
70 – 2(5 + 3)
70 – 2(8)
70 – 16
54 correct
Order of Operations is not
an isolated skill.
This skill applies to almost every topic in Math.
Remember that “Aunt Sally” is used with
evaluating formulas, solving equations, evaluating
algebraic expressions, simplifying monomials &
polynomials, etc…
Order of Operations
Get your pencil and calculator ready and try
these problems.
1) 20 + 3(5 – 1) =
6) 48 / 3 + 5 =
2) 3 + 2²(1 + 8) =
7) 3(6 + 4)(5 – 3) =
3) (5 · 4)² =
8) 100 – 4(7 – 4)³ =
4) 2(3 + 5) – 9 =
9) 1² + 2³ + 3³ =
5) 2[13 – (1 + 6)] =
10) (24 – 6)/2 =
Here is an example of an algebraic expression.
4 is the coefficient.
X is the variable.
7 is the constant.
Algebraic Expression
4x + 7
4 = coefficient
x = variable
7 = constant
Evaluating Algebraic Expressions
To evaluate an algebraic expression, substitute a
given number for the variable, and find the value
of the resulting numerical expression.
X – 5 for x = 12
(12) – 5 = 7
2y + 1 for y = 4
2(4) + 1 = 9
6(n + 2) – 4 for n = 5, 6, 7
38, 44, 50
Evaluate each expression for
t = 0, x = 1.5, y = 6, z = 23
1) y + 5
6) 3(4 + y)
2) 3z – 3y
7) 3(6 + t) – 1
3) z – 2x
8) 2(y – 6) + 3
4) xy
9) y(4 + t) - 5
5) 4(y – x)
10) x + y + z
Properties of Real Numbers
Properties of Real Numbers
Property
Example
1
Commutative Property of Addition
a+b=b+a
2+3=3+2
2
Commutative Property of Multiplication
a·b=b·a
2 · (3) = 3 · (2)
3
Associative Property of Addition
a + (b + c) = (a + b) + c
2 + (3 + 4) = 2 + (3 + 4)
4
Associative Property of Multiplication
a · (b · c) = (a · b) · c
2 · (3 · 4) = (2 · 3) · 4
5
Distributive Property
a · (b · c) = a · b + a · c
2 · (3 + 4) = 2 · 3 + 2 · 4
6
Identity Property of Addition
a+0=a
3+0=3
7
Identity Property of Multiplication
a·1=a
3·1=3
8
Additive Inverse Property
a + (-a) = 0
3 + (-3) = 0
9
Multiplicative Inverse Property
a · (1/a) = 1
3 · (1/3) = 1
10
Property of Zero
a·0=0
5·0=0