1. Chapter 2 Equations and Inequalities in One Variable
Section 1
Linear Equations: The Addition and Multiplication Properties of Equality

2. Section 2.1 Objectives
1 Determine Whether a Number Is a Solution of an Equation
2 Use the Addition Property of Equality to Solve Linear Equations
3 Use the Multiplication Property of Equality to Solve Linear Equations

3. Linear Equations in One Variable
A linear equation in one variable is an equation that can be written in the form ax + b = c where a, b, and c are real numbers and a  0.
3x + 5 = 25
The expressions are called the sides of the equation.

4. Solutions
The solution of a linear equation is the value or values of the variable that make the equation a true statement. The set of all solutions of an equation is called the solution set. The solution satisfies the equation.
Example:
Determine if x = –1 is a solution to the equation.
–3(x – 3) = –4x + 3 – 5x
–3[(–1) – 3] = – 4(–1) + 3 – 5(–1)
– 3(–4) =
12 = 12
True. x = – 1 is a solution

5. Solving Equations
Linear equations are solved by writing a series of steps that result in the equation
x = a number
One method for solving equations is to write a series of equivalent equations.
Two or more equations that have precisely the same solutions are called equivalent equations.
3 + 5 = 8
1 + 7 = 2 + 6

6. Solution to the equation.
Addition Property
The Addition Property of Equality states that for real numbers a, b, and c,
if a = b, then a + c = b + c.
We need to find the value of y.
6 + y = 11
6 + y + (6) = 11 + (6)
Adding (6) to both sides of the equation will maintain the balance of the equation.
y = 5
Left side
Right side 6
Solution to the equation.

7. Using the Addition Property
Example:
Solve the linear equation x  9 = 22.
Step 1: Isolate the variable x on the left side of the equation.
x  =
Add 9 to both sides of the equation.
Step 2: Simplify the left and right sides of the equation.
x = 31
Apply the Additive Inverse Property.
Step 3: Check to verify the solution.
x  9 = 22
31  9 = 22
22 = 22 

8. Multiplication Property
The Multiplication Property of Equality states that for real numbers a, b, and c, where c  0,
if a = b, then ac = bc.
We need to find the value of x.
Multiplying both sides of the equation by will maintain the balance of the equation.
x = 28
Right side
Left side
× 7
Solution to the equation.

9. Using the Multiplication Property
Example:
Solve the linear equation 3x = 81
Step 1: Get the coefficient of the variable x to be 1.
(3x) = (81)
Multiply each side of the equation by
Step 2: Simplify the left and right sides of the equation.
x = 27
Apply the Multiplicative Inverse Property.
Step 3: Check to verify the solution.
3x = 81
3(27) = 81
81 = 81 