2.1 The Addition Property of Equality

Definitions. An equation is a statement asserting that two algebraic expressions are equal. A solution of an equation is a number that makes the equation true when it replaces the variable. An equation is solved by finding its solution set, the set of all solutions. Equations with exactly the same solution sets are equivalent equations. Equation Remember that an equation (to solve) includes an equals symbol which distinguishes is from an expression (to simplify or evaluate). Expression Slide 2.1-3

Identify linear equations. Objective 1 Identify linear equations. Slide 2.1-4

Linear Equation in One Variable Identify linear equations. The simplest type of equation is a linear equation. Linear Equation in One Variable A linear equation in one variable can be written in the form where A, B, and C are real numbers, and with A ≠ 0. Linear Equations Nonlinear Equations Slide 2.1-5

Use the addition property of equality. Objective 2 Use the addition property of equality. Slide 2.1-6

Addition Property of Equality Use the addition property of equality. To solve an equation, add the same number to each side. The justifies this step. Addition Property of Equality If A, B, and C are real numbers, then the equations and are equivalent equations. That is, we can add the same number to each side of an equation without changing the solution. Equations can be thought of in terms of a balance. Thus, adding the same quantity to each side does not affect the balance. Slide 2.1-7

Applying the Addition Property of Equality CLASSROOM EXAMPLE 1 Applying the Addition Property of Equality Solve. Solution: Check: The solution set is The final line of the check does not give the solution to the problem, only a confirmation that the solution found is correct. Do NOT write the solution set as {x = 9}. This is incorrect notation. Simply write {9}. Slide 2.1-8

Applying the Addition Property of Equality CLASSROOM EXAMPLE 2 Applying the Addition Property of Equality Solve. Solution: Check: The solution set is Slide 2.1-9

Applying the Addition Property of Equality CLASSROOM EXAMPLE 3 Applying the Addition Property of Equality Solve. Solution: Check: The solution set is Slide 2.1-11

Subtracting a Variable Expression CLASSROOM EXAMPLE 4 Subtracting a Variable Expression Solve. Solution: Check: The solution set is Slide 2.1-12

Use the addition property of equality. (cont’d) The addition property of equality says that the same number may be added to each side of an equation. In Section 1.5, subtraction was defined as addition of the opposite. Thus, we can also use the following rule when solving an equation. The same number may be subtracted from each side of an equation without changing the solution. Slide 2.1-10

Applying the Addition Property of Equality Twice CLASSROOM EXAMPLE 5 Applying the Addition Property of Equality Twice Solve. Solution: Check: The solution set is Slide 2.1-13

Simplify, and then use the addition property of equality. Objective 3 Simplify, and then use the addition property of equality. Slide 2.1-14

Combining Like Terms When Solving CLASSROOM EXAMPLE 6 Combining Like Terms When Solving Solve. Solution: Check: The solution set is Slide 2.1-15

Using the Distributive Property When Solving CLASSROOM EXAMPLE 7 Using the Distributive Property When Solving Solve. Solution: Check: The solution set is Be careful to apply the distributive property correctly, or a sign error may result. Slide 2.1-16