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
lw
lw
V
Vlw
V
lw
V  lwh