Unit 1.4 Introduction to mathematics Frank Budnick

1. Introduction to Mathematics
Week 1
Review of Basic Arithmetic Operations & Algebra
• Unit 1.4
Expressions, Equations & Their Properties

2. Expressions
• When we combine numbers and variables in a valid way, using operations
such as addition, subtraction, multiplication, division, exponentiation, and
other operations and functions as yet unlearned, the resulting combination
of mathematical symbols is called a mathematical expression.
• Expressions are general little statements that do NOT have an equals sign.
For example
• 2𝑥 − 4
• 2𝑚3
+ 12𝑛
• Expressions may be stated in terms of one or more variables. In the above
examples, x, m and n are the variables.

3. Equations
• An equation states the equality of two algebraic expressions.
• The algebraic expressions maybe stated in terms of one or more variables.
For example,
• 3𝑥 − 10 = 22 − 5𝑥
•
2𝑟−5𝑠+8𝑡
3
= 100
• The term variable is used because various numbers can be substituted for
the letters.

4. Solution of an Equation
• The solution of an equation consist of those numbers which when
substituted for the variables make the equation true. The numbers, or values
of the variables, which make the equation true are referred to as he roots of
the equation. We say that the roots are the values of the variables which
satisfy the equation. For example, if we have the equation
3𝑥 − 10 = 22 − 5𝑥.
• If we substitute the number 0 in place of x, the result is
−10 = 22
• Which is untrue. The value x = 0 is NOT the root of the equation. However,
substituting the number 4 in place of x, gives us.
• 3 4 − 10 = 22 − 5 4
• 2 = 2
• Hence x = 4 is the root of the equation.

5. Solving an equation refers to the process of finding the roots of an equation, if
any roots exits. In order to solve equations, we normally must manipulate or
rearrange them. The following rules indicate allowable operations.

6. Types of Equations
• Identity
An identity is an equation which is true for all values of the variables. For
example,
6𝑥 + 12 =
12𝑥 + 24
2
In such types of equations, any values assigned to the variables will make both
sides of the equation equal.
• Conditional Equation
This equation is true for only a limited number of values of the variable. For
example,
𝑥 + 1 = 4
Is only true, when x = 3.

7. • False Statement
A false statement or a contradiction, is an equation that is never true. There are
no values of the variables that make the equation true. For example,
𝑥 = 2 + 𝑥
We indicate that the two sides are not equal by using the symbol ≠, like this.
𝑥 ≠ 2 + 𝑥

8. Thank you