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Section 1.3 Models and Applications Problem Solving with Linear Equations Strategy for Solving Word Problems Step 1: Read the problem carefully. Attempt to state the problem in your own words and state what the problem is looking for. Let any variable represent one of the quantities in the problem. Step2: If necessary, write expressions for any other unknown quantities in the problem in terms of x. Step 3: Write an equation in x that models the verbal conditions of the problem. Step 4: Solve the equation and answer the problem’s question. Step 5: Check the solution in the original wording of the problem, not in the equation obtained from the words. Example Spice Drops candy calorie count exceeds Smarties candy calorie count by 70 calories per serving. If the sum of one serving of each candy equals 170 calories find the calorie count of each kind of candy. Step 1: Represent one of the quantities Step 2: Represent the other quantity. Step 3:Write an equation in x that models the conditions. Step 4: Solve the equation and answer the question. Step 5: check the proposed solution. Example The percentage of women in the labor force and the percentage of men in the labor force is illustrated in the graph at left. The decrease yearly of men in the labor force is ¼% and the increase in women in the labor force is ½%. If there are presently 70 million men and 60 million women in the labor force, when will the number of both sexes be equal? Graphing Calculator Solving the previous problem using intersection. Let y1 be the left side of the equation and y2 be the right side of the equation. y 2 60 .005 60 x y1=70-.0025 70 x Example A woman who was going to retire had $100,000 that she invested in her local bank. She put some of the money in a money market account at 3 ½% and some in a certificate of deposit at 4%. If the first year’s interest is $3850, how much money did she put in each account? % Money market CD amount Interest Example A local telephone company charges $11 for local phone service and an additional $ .10 for each long distance phone call. A second local telephone company charges $14 for local service and an additional $ .05 for each long distance phone call. For how many minutes of long-distance calls will the costs for the two companies be the same? Example In 2002, the median annual income for people with an advanced college degree was $73,000. This is a 170% increase over the median income in 1982 of people with an advanced degree. What were people with an advanced college degree making in 1982? Solving a Formula for One of Its Variables The formula for the perimeter of a rectangle is given below. Solve for the length of the rectangle. Example Solve for r : C 2r The formula below describes the amount A, that a principal of P dollars is worth after t years when invested at a simple annual interest rate, r. Solve this formula for P. Example Solve for h: A hb 1 2 1 1 2 hb2 If you invest a total of $20,000 in two accounts. One account pays 5% and another account pays 3%. If you make $900 in interest the first year, how much did you invest at each percent? (a) $15,500- 5%, $4,500 -3% (b)$12,000-5%, $8,000-3% (c)$15,000-5%, $5,000-3% (d)$10,000-5%, $10,000-3% Solve for h. (a) (b) (c) (d) V lw lw V Vlw V lw V lwh