Sullivan Algebra and Trigonometry: Section 1.1 Linear Equations Objectives of this Section Solve a Linear Equation Solve Equations That Lead to Linear Equations Solve Applied Problems Involving Linear Equations

The set of all values of a variable that make an equation a true statement are called solutions, or roots, of the equation. Example: The statement x + 5 = 9 is true when x = 4 and false for any other choice of x. Thus, 4 is a solution of the equation. We also say that 4 satisfies the equation x + 5 = 9.

We solve equations by creating a series of equivalent equations, that is equations that have precisely the same solution set. Example: The equations 3x – 4 = 5x and 2x = -4 are equivalent equations, since the solution set for both equation is {-2}.

Procedures that Result in Equivalent Equations 1. Interchange the two sides of the equation. 2. Simplify the sides of the equation by combining like terms, eliminating parenthesis, and so on. 3. Add or subtract the same expression on both sides of the equation. 4. Multiply or divide both sides of the equation by the same nonzero expression. 5. If one side of the equation is 0 and the other side can be factored, then set each factor equal to 0.

Solve: Example

Solve:

Example Solve the equation:  xxxx  3124

A linear equation in one variable is equivalent to an equation of the form: Note that the previous example,  xxxx  3124 Is an example of a linear equation, since it is equivalent to: Both equations have the same solution set: { }

Example

Steps for Setting Up Applied Problems Step 1: Read the problem carefully, perhaps two or three times. Identify what you are looking for. Step 2: Assign a letter (variable) to represent what you are looking for. Express any remaining unknown quantities in terms of that variable. Step 3: Make a list of known facts and translate them into mathematical expressions. These may take the form of an equation or later, an inequality. Step 4: Solve the equation for the variable and answer the question using a sentence. Step 5: Check your answer with the facts of the problem.

Example: Yolanda, Mary, and Sophie won $200,000 playing the lottery. Based on how much each contributed to buy the ticket, Mary gets four fifths of what Yolanda gets, while Sophie gets three fourth of what Mary gets. How much does each receive?