1. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1
Equations and
Inequalities
1.3 Models and Applications
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

2. Objectives:
Use linear equations to solve problems.
Solve a formula for a variable.

3. Problem Solving with Linear Equations
A model is a mathematical representation of a real-world
situation. We will use the following general steps in
solving word problems:
Step 1 Let x represent one of the unknown quantities.
Step 2 Represent other unknown quantities in terms of x.
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 in the original
wording of the problem.

4. Example: Using Linear Equations to Solve Problems
The median starting salary of a computer science major
exceeds that of an education major by $21 thousand. The
median starting salary of an economics major exceeds
that of an education major by $14 thousand. Combined,
their median starting salaries are $140 thousand. Determine
the median starting salaries of education majors, computer
science majors, and economics majors with bachelor’s
degrees.

5. Example: Using Linear Equations to Solve Problems (continued)
Step 1 Let x represent one of the unknown quantities.
x = median starting salary of an education major
Step 2 Represent other unknown quantities in terms of x.
x + 21 = median starting salary of a computer science major
x + 14 = median starting salary of an economics major
Step 3 Write an equation in x that models the conditions.

6. Example: Using Linear Equations to Solve Problems (continued)
Step 4 Solve the equation and answer the question.
starting salary of an education major is x = 35
starting salary of a computer science major is x + 21 = 56
starting salary of an economics major is x + 14 = 49

7. Example: Using Linear Equations to Solve Problems (continued)
Step 4 Solve the equation and answer the question.
(continued)
The median starting salary of an education major is
$35 thousand, the median starting salary of a computer
science major is $56 thousand, and the median starting
salary of an economics major is $49 thousand.

8. Example: Using Linear Equations to Solve Problems (continued)
Step 5 Check the proposed solution in the original
wording of the problem.
The problem states that combined, the median starting
salaries are $140 thousand. Using the median salaries we
determined in Step 4, the sum is
$35 thousand + $56 thousand + $49 thousand,
or $140 thousand, which verifies the problem’s conditions.

9. Example: Using Linear Equations to Solve Problems
You are choosing between two texting plans. Plan A has a
monthly fee of $15 with a charge of $0.08 per text. Plan B
has a monthly fee of $3 with a charge of $0.12 per text. For
how many text messages will the costs for the two plans be
the same?
Step 1 Let x represent one of the unknown quantities.
x = charge per text
Step 2 Represent other unknown quantities in terms of x.
Plan A = 0.08x +15
Plan B = 0.12x + 3

10. Example: Using Linear Equations to Solve Problems (continued)
Step 3 Write an equation in x that models the conditions.
Step 4 Solve the equation and answer the question.
The cost of the two plans will be the same for
300 text messages.

11. Example: Using Linear Equations to Solve Problems (continued)
Step 5 Check the proposed solution in the original
wording of the problem.
The problem asks for how many text messages will the
two plans be the same?
The cost of Plan A for 300 text messages is
0.08(300) + 15 = = 39
The cost of Plan B for 300 text messages is
0.12(300) + 3 = = 39
For 300 text messages, the plan costs are equal,
which verifies the problem’s conditions.

12. Example: Using Linear Equations to Solve Problems
After a 30% price reduction, you purchase a new computer
for $840. What was the computer’s price before the
reduction?
Step 1 Let x represent one of the unknown quantities.
x = price before reduction
Step 2 Represent other unknown quantities in terms of x.
price of new computer = x – 0.3x
Step 3 Write an equation in x that models the conditions.

13. Example: Using Linear Equations to Solve Problems (continued)
Step 4 Solve the equation and answer the question.
The price of the computer before the reduction was $1200.
Step 5 Check the proposed solution in the original
wording of the problem.
The price before the reduction, $1200, minus the 30%
reduction should equal the reduced price given in the original
wording, $840.

14. Example: Using Linear Equations to Solve Problems (continued)
Step 5 Check the proposed solution in the original
wording of the problem. (continued)
This verifies that the computer’s price before the reduction
was $1200.

15. Example: Using Linear Equations to Solve Problems
You inherited $5000 with the stipulation that for the first year
the money had to be invested in two funds paying 9% and
11% annual interest. How much did you invest at each rate
if the total interest earned for the year was $487?
Step 1 Let x represent one of the unknown quantities.
x = the amount invested at 9%
Step 2 Represent other unknown quantities in terms of x.
5000 – x = the amount invested at 11%
Step 3 Write an equation in x that models the conditions.

16. Example: Using Linear Equations to Solve Problems (continued)
Step 4 Solve the equation and answer the question.
the amount invested at 9% = x = 3150
the amount invested at 11% = 5000 – x = 1850
You should invest $3150 at 9% and $1850 at 11%.

17. Example: Using Linear Equations to Solve Problems (continued)
Step 5 Check the proposed solution in the original
wording of the problem.
The problem states that the total interest from the dual
investments should be $487.
The interest earned at 9% is 0.09(3150) =
The interest earned at 11% is 0.11(1850) =
The total interest is = This
verifies that the amount invested at 9% should be
$3150 and that the amount invested at 11% should
be $1850.

18. Example: Using Linear Equations to Solve Problems
The length of a rectangular basketball court is 44 feet more
than the width. If the perimeter of the basketball court is
288 feet, what are its dimensions?
Step 1 Let x represent one of the unknown quantities.
x = the width of the basketball court
Step 2 Represent other unknown quantities in terms of x.
x + 44 = the length of the basketball court
Step 3 Write an equation in x that models the conditions.

19. Example: Using Linear Equations to Solve Problems (continued)
Step 4 Solve the equation and answer the question.
the width of the basketball court is x = 50 ft
the length of the basketball court is x + 44 = 94 ft
The dimensions of the basketball court are 50 ft by 94 ft

20. Example: Using Linear Equations to Solve Problems (continued)
Step 5 Check the proposed solution in the original
wording of the problem.
The problem states that the perimeter of the basketball court
is 288 feet. If the dimensions are 50 ft by 94 ft, then the
perimeter is 2(50) + 2(94) = = This
verifies the conditions of the problem.

21. Solving a Formula For A Variable
Solving a formula for a variable means rewriting the formula so that the variable is isolated on one side of the equation.

22. Example: Solving a Formula for a Variable

23. Example: Solving a Formula for a Variable That Occurs Twice