Section 2.4 Applications of Linear Equations: Introduction to Problem Solving 147 1. Problem-Solving Strategies Linear equations can be used to solve many real-world applications. However, with “word problems,” students often do not know where to start. To help organize the problem-solving process, we offer the following guidelines: Problem-Solving Flowchart for Word Problems • Familiarize yourself with the problem. Identify the unknown, and if possible, estimate the answer. Step 1 Read the problem carefully. • Identify the unknown quantity or quantities. Let x or another variable represent one of the unknowns. Draw a picture and write down relevant formulas when appropriate. Step 2 Assign labels to unknown quantities. Step 3 Write a verbal model. • Write an equation in words. • Replace the verbal model with a mathematical equation using x or another variable. Step 4 Write a mathematical equation. • Solve for the variable using the steps for solving linear equations. Step 5 Solve the equation. • Once you have obtained a numerical value for the variable, recall what it represents in the context of the problem. Can this value be used to determine other unknowns in the problem? Write an answer to the word problem in words. Interpret the results and write the final answer in words. Step 6 2. Translating to Linear Equations We have already practiced translating an English sentence to a mathematical equation. Recall from Section 1.2 that several key words translate to the algebraic operations of addition, subtraction, multiplication, and division. Translating to a Linear Equation Example 1 The sum of a number and negative eleven is negative fifteen. Find the number. Solution: Concepts 1. Problem-Solving Strategies 2. Translating to Linear Equations 3. Consecutive Integer Problems 4. Applications of Linear Equations Instructor Note: Remind students that x is a convenient variable, but any letter can be used. For example, for an unknown width use w. Applications of Linear Equations: Introduction to Problem Solving Section 2.4 Classroom Example: p. 153, Exercise 10 Answer 1. The number is 19. Step 1: Read the problem. Let x represent the unknown number. Step 2: Label the unknown. the sum of is Step 3: Write a verbal model. Step 4: Write an equation. Step 5: Solve the equation. 1a number2 1112 1152 x 1112 15 x 1112 11 15 11 x 4 The number is 4. Step 6: Write the final answer in words. Avoiding Mistakes Once you have reached a solution to a word problem, verify that it is reasonable in the context of the problem. Skill Practice 1. The sum of a number and negative seven is 12. Find the number.
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