878 CHAPTER 9 Systems and Matrices Application of Systems of Equations Many applied problems involve more than one unknown quantity. Although some problems with two unknowns can be solved using just one variable, it is often easier to use two variables. To solve a problem with two unknowns, we must write two equations that relate the unknown quantities. The system formed by the pair of equations can then be solved using the methods of this chapter. The following steps, based on the six-step problem-solving method introduced earlier, give a strategy for solving such applied problems. Solving an Applied Problem by Writing a System of Equations Step 1 Read the problem carefully. What information is given? What is to be found? Step 2 Assign variables to represent the unknown values. Write down what each variable represents. Make a sketch, diagram, or table, as needed. Step 3 Write a system of equations that relates the unknowns. Step 4 Solve the system of equations. Step 5 State the answer. Label it appropriately. Does it seem reasonable? Step 6 Check the answer in the words of the original problem. EXAMPLE 5 Using a Linear System to Solve an Application Salaries for the same position can vary depending on the location. In 2019, the average of the median salaries for the position of Accountant I in San Diego, California, and Salt Lake City, Utah, was $53,763.50. The median salary in San Diego, however, exceeded the median salary in Salt Lake City by $6733. Determine the median salary for the Accountant I position in San Diego and in Salt Lake City. (Data from www.salary.com) SOLUTION Step 1 Read the problem. We must find the median salary of the Accountant I position in San Diego and in Salt Lake City. Step 2 Assign variables. Let x represent the median salary of the Accountant I position in San Diego and y represent the median salary for the same position in Salt Lake City. Step 3 Write a system of equations. Since the average of the two medians for the Accountant I position in San Diego and Salt Lake City was $53,763.50, one equation is as follows. x + y 2 = 53,763.50 Multiply each side of this equation by 2 to clear the fraction and obtain an equivalent equation. x + y = 107,527 (1) The median salary in San Diego exceeded the median salary in Salt Lake City by $6733. Thus, x - y = 6733, which gives the following system of equations. x + y = 107,527 (1) x - y = 6733 (2)
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