This document covers the concepts and properties of linear equations, including how to identify and solve them. It explains the characteristics of identities, conditional equations, and contradictions, along with methods for solving literal equations. Additionally, the document includes examples and assignments for further practice.

1. 1.1 Linear Equations
Chapter 1 Equations and Inequalities

2. Concepts and Objectives
⚫ Identify basic properties of equations
⚫ Solve linear equations
⚫ Identify equations as identities, conditional equations,
and contradictions
⚫ Solve for a specific variable (literal equations)

3. Basic Properties of Equations
⚫ An equation is a statement that two expressions are
equal. (Ex.: x + 2 = 9 or 7y +1 = 5y + 23)
⚫ To solve an equation means to find all numbers that
make the equation a true statement. These numbers are
called solutions or roots of the equation. The set of all
solutions to an equation is called its solution set.
⚫ We can use the addition and multiplication
properties of equality to help us solve equations.

4. Basic Properties of Equations
⚫ Addition (and Subtraction) Property of Equality:
⚫ Multiplication (and Division) Property of Equality:
For real numbers a, b, and c:
If a = b, then ac = bc
If a = b and c  0, then
a b
c c
=
For real numbers a, b, and c:
If a = b, then a + c = b + c
If a = b, then a ‒ c = b ‒c

5. Linear Equations
⚫ A linear equation in one variable is an equation that
can be written in the form
ax + b = 0,
where a and b are real numbers and a  0.
⚫ A linear equation is also called a first-degree equation
since the degree of the variable is either one or zero.
⚫ To solve a linear equation, use the properties of equality
to isolate the variable on one side and the solution on
the other.

6. Linear Equations (cont.)
⚫ If solving a linear equation leads to
⚫ a true statement such as 0 = 0, the equation is an
identity. Its solution set is  or {all real numbers}.
⚫ a single solution such as x = 3, the equation is
conditional. Its solution set consists of a single
element.
⚫ a false statement such as 0 = 7, the equation is a
contradiction. Its solution set is  or { }.

7. Linear Equations (cont.)
Example: Decide whether each equation is an identity,
conditional equation, or a contradiction. Give the solution
set.
⚫ 3x + 2 ‒ x = 2x + 6
⚫ 5x ‒ 4 = 11
⚫ 2x +5 + x = 3x + 9

8. Linear Equations (cont.)
Example: Decide whether each equation is an identity,
conditional equation, or a contradiction. Give the solution
set.
⚫ 3x + 2 ‒ x = 2x + 6
3x + 6 ‒ x = 2x + 6
2x + 6 = 2x + 6
0 = 0
⚫ 5x ‒ 4 = 11
⚫ 2x +5 + x = 3x + 9

9. Linear Equations (cont.)
Example: Decide whether each equation is an identity,
conditional equation, or a contradiction. Give the solution
set.
⚫ 3x + 2 ‒ x = 2x + 6 identity 
3x + 6 ‒ x = 2x + 6
2x + 6 = 2x + 6
0 = 0
⚫ 5x ‒ 4 = 11
⚫ 2x +5 + x = 3x + 9

10. Linear Equations (cont.)
Example: Decide whether each equation is an identity,
conditional equation, or a contradiction. Give the solution
set.
⚫ 3x + 2 ‒ x = 2x + 6 identity 
⚫ 5x ‒ 4 = 11
5x = 15
x = 3
⚫ 2x +5 + x = 3x + 9

11. Linear Equations (cont.)
Example: Decide whether each equation is an identity,
conditional equation, or a contradiction. Give the solution
set.
⚫ 3x + 2 ‒ x = 2x + 6 identity 
⚫ 5x ‒ 4 = 11 conditional {3}
5x = 15
x = 3
⚫ 2x +5 + x = 3x + 9

12. Linear Equations (cont.)
Example: Decide whether each equation is an identity,
conditional equation, or a contradiction. Give the solution
set.
⚫ 3x + 2 ‒ x = 2x + 6 identity 
⚫ 5x ‒ 4 = 11 conditional {3}
⚫ 2x +5 + x = 3x + 9
3x + 5 = 3x + 9
0 = 4

13. Linear Equations (cont.)
Example: Decide whether each equation is an identity,
conditional equation, or a contradiction. Give the solution
set.
⚫ 3x + 2 ‒ x = 2x + 6 identity 
⚫ 5x ‒ 4 = 11 conditional {3}
⚫ 2x +5 + x = 3x + 9 contradiction 
3x + 5 = 3x + 9
0 = 4

14. Literal Equations
⚫ A literal equation is an equation that relates two or
more variables. Formulas are examples of literal
equations.
⚫ When using literal equations, sometimes the variable is
not the one that is isolated, and so we have to solve for a
different variable.
⚫ In this case we use some of the same methods used to
solve other equations – we treat the specified
variable as if it were the only variable, and the other
variables we treat as constants (although we
generally can’t combine them).

15. Literal Equations (cont.)
Examples: Solve for the specified variable.
(1) , for t (2) , for kI Prt= 2
S kr krm= +

16. Literal Equations (cont.)
Examples: Solve for the specified variable.
(1) , for t (2) , for k
I Pr
Pr P
t
r
I
t
Pr
=
=
I Prt= 2
S kr krm= +

17. Literal Equations (cont.)
Examples: Solve for the specified variable.
(1) , for t (2) , for k
I Pr
Pr P
t
r
I
t
Pr
=
=
I Prt= 2
S kr krm= +
( )
( )
2
2
2 2
2
S r rm
k r rmS
r rm r rm
S
k
r rm
k= +
+
=
+ +
=
+

18. Classwork
⚫ 1.1 Assignment (College Algebra)
⚫ Page 88: 10-28 (even); page 71: 64-92 (×4);
page 61: 76-96 (×4)
⚫ 1.1 Classwork Check
⚫ Quiz 0.7