Section 4.4 Problem Solving Using Systems of Equations

Objectives Assign variables to two unknowns Use systems to solve geometry problems Use systems to solve number-value problems Use systems to solve interest, uniform motion, and mixture problems

In previous chapters, many applied problems were modeled and solved with an equation in one variable. In this section, the application problems involve two unknowns.  It is often easier to solve such problems using a two-variable approach. Objective 1: Assign Variables to Two Unknowns

The following steps are helpful when solving problems involving two unknown quantities.  Problem-Solving Strategy: 1. Analyze the problem by reading it carefully to understand the given facts. Often a diagram or table will help you visualize the facts of the problem. 2. Assign variables to represent unknown values in the problem. This means, in most cases, to let x = one of the unknowns that you are asked to find, and y = the other unknown. 3. Solve the system of equations using graphing, substitution, or elimination. 4. State the conclusion clearly. Be sure to include the units. 5. Check the results in the words of the problem, not the equations that were formed in step 3. Objective 1: Assign Variables to Two Unknowns

EXAMPLE 1 Each year, Academy Award winners are presented with Oscars. The 13.5-inch statuette has a base on which a gold- plated figure stands. The figure itself is 7.5 inches taller than its base. Find the height of the figure and the height of the base. Motion Pictures

Two angles are said to be complementary if the sum of their measures is 90°. Two angles are said to be supplementary if the sum of their measures is 180°. Objective 2: Use Systems to Solve Geometry Problems

EXAMPLE 3 In 1917, James Montgomery Flagg created the classic I Want You poster to help recruiting for World War I. The perimeter of the poster is 114 inches, and its length is 9 inches less than twice its width. Find the length and the width of the poster. History Analyze The perimeter of the rectangular poster is 114 inches. The length is 9 inches less than twice the width. Find the length and the width of the poster.

Solve Example 4 on the next few slides. Objective 3: Use Systems to Solve Number-Value Problems

EXAMPLE 4 At a school, two picture packages are available, as shown in the illustration. Find the cost of a class picture and the cost of an individual wallet-size picture. Photography

Solve Example 6 on the next few slides. Objective 4: Use Systems to Solve Interest, Uniform Motion, and Mixture Problems

EXAMPLE 6 A boat traveled 30 miles downstream in 3 hours and made the return trip in 5 hours. Find the speed of the boat in still water and the speed of the current. Boating