Section 1.3Models 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. 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.

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

ExampleThe 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.

1=70-.0025 70 xy 2 60 .005 60 xy

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. Solve for the length of the rectangle.

Example

Solve for r : 2C 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:1 1

1 22 2

A hb hb

(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%

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)

(b)

(c)

(d)

V lwhSolve for h.

V

l wlw

VVlw

V

lw