Copyright © Cengage Learning. All rights reserved. Systems of Linear Equations and Inequalities in Two Variables 7

Copyright © Cengage Learning. All rights reserved. Section 7.4 Solving Applications of Systems of Linear Equations

3 Objective Solve an application using a system of linear equations. 1 1

4 Solve an application using a system of linear equations 1.

5 Solve an application using a system of linear equations The following steps are helpful when solving applications involving two unknown quantities. Problem Solving 1. Read the problem and analyze the facts. Identify the unknowns by asking yourself “What am I asked to find?” a. Select different variables to represent two unknown quantities. b. Write a sentence to define each variable.

6 Solve an application using a system of linear equations 2. Form two equations involving each of the two variables. This will create a system of two equations in two variables. (This may require reading the problem several times to understand the given facts. What information is given? Is there a formula that applies to this situation? Will a sketch, chart, or diagram help you visualize the facts of the problem?)

7 Solve an application using a system of linear equations 3. Solve the system using the most convenient method: graphing, substitution, or elimination. 4. Check the solution in the words of the problem.

8 Example – Farming A farmer raises wheat and soybeans on 215 acres. If he wants to plant 31 more acres in wheat than in soybeans, how many acres of each should he plant? 1.What Am I asked to Find? The farmer plants two fields, one in wheat and one in soybeans. We are asked to find how many acres of each he should plant. So, we let w represent the number of acres of wheat and s represent the number of acres of soybeans.

9 Example – Farming 2.Form two equations We know that the number of acres of wheat planted plus the number of acres of soybeans planted will equal a total of 215 acres. So we can form the equation Since the farmer wants to plant 31 more acres in wheat than in soybeans, we can form the equation cont’d

10 Example – Farming 3.Solve the system We can now solve the system (1)w + s = 215 (2)w – s = 31 by the elimination method. w = 123 Divide both sides by 2. cont’d

11 Example – Farming To find s, we substitute 123 for w in Equation 1. w + s = s = 215 s = 92 State the conclusion The farmer should plant 123 acres of wheat and 92 acres of soybeans. Substitute 123 for w. Subtract 123 from both sides. cont’d

12 Example – Farming 4.Check the result The total acreage planted is , or 215 acres. The area planted in wheat is 31 acres greater than that planted in soybeans, because 123 – 92 = 31. The answers check. cont’d