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.

Solving word problems with the word exceeds can be tricky Solving word problems with the word exceeds can be tricky. It is helpful to identify the smaller quantity as x, then add to it to represent the larger quantity. If Tim’s height exceeds Tom’s by y inches then Tom is shorter, x, and Tim’s height is x+y.

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?

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? % amount Interest Money market CD

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 The formula for the perimeter of a rectangle is given below. Solve for the length of the rectangle.

Example Solve for r :

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:

Practice Problems 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)