1. Algebra: Equations and Inequalities
CHAPTER 6
Algebra: Equations and Inequalities

2. Applications of Linear Equations
6.3
Applications of Linear Equations

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

4. Strategy for Solving Word Problems
Step 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable) represent one of the quantities in the problem. Step 2 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.

5. Algebraic Translations of English Phrases
Addition
Subtraction
Multiplication
Division
sum
more than
increased by
minus
decreased by
subtracted from
difference between
less than
fewer than
times
product of
percent of a number
multiplied by
twice
divided by
quotient
reciprocal

6. Example: Education Pays Off
The bar graph shows the ten most popular college majors with median, or middlemost, starting salaries for recent college graduates.

7. Example: Education Pays Off
The median starting salary of a business major exceeds that of a psychology major by $8 thousand. The median starting salary of an English major exceeds that of a psychology major by $3 thousand. Combined, their median starting salaries are $116 thousand. Determine the median starting salaries of psychology majors, business majors, and English majors with bachelor’s degrees.

8. Example 1: continued
Step 1 Let x represent one of the unknown quantities. We know something about the median starting salaries of business majors and English majors: Business majors earn $8 thousand more than psychology majors and English majors earn $3 thousand more than psychology majors. We will let x = the median starting salary, in thousands of dollars, of psychology majors. x + 8 = the median starting salary, in thousands of dollars, of business majors. x + 3 = the median starting salary, in thousands of dollars, of English majors.

9. Example: continued
Step 3: Write an equation in x that models the conditions. x + (x + 8) + (x + 3) = 116 Step 4: Solve the equation and answer the question.

10. Example: continued
starting salary of psychology majors: x = 35 starting salary of business majors: x + 8 = = 43 starting salary of English majors: x + 3 = = 38. Step 5: Check the proposed solution in the wording of the problem. The solution checks.

11. Example: Selecting Monthly Text Message Plan
You are choosing between two texting plans. Plan A has a monthly fee of $20.00 with a charge of $0.05 per text. Plan B has a monthly fee of $5.00 with a charge of $0.10 per text. Both plans include photo and video texts. 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. Let x the number of text messages for which the two plans cost the same.

12. Example: continued
Step 2 Represent other unknown quantities in terms of x.
There are no other unknown quantities, so we can skip this step.
Step 3 Write an equation in x that models the conditions.
The monthly cost for plan A is the monthly fee, $20.00, plus the per-text charge, $0.05, times the number of text messages, x. The monthly cost for plan B is the monthly fee, $5.00, plus the per-text charge, $0.10, times the number of text messages, x.

13. Example: continued
Step 4 Solve the equation and answer the question. Because x represents the number of text messages for which the two plans cost the same, the costs will be the same for 300 texts per month.

14. Example: continued
Step 5 Check the proposed solution in the original wording of the problem. Cost for plan A = $20 + $0.05(300) = $20 + $15 = $35 Cost for plan B = $5 + $0.10(300) = $5 + $30 = $35. With 300 text messages, both plans cost $35 for the month. Thus, the proposed solution, 300 text messages, satisfies the problem’s conditions.

15. Example: A Price Reduction on a Digital Camera
Your local computer store is having a terrific sale on digital cameras. After a 40% price reduction, you purchase a digital camera for $276. What was the camera’s price before the reduction? Step 1 Let x represent one of the unknown quantities. We will let x = the original price of the digital camera prior to the reduction. Step 2 Represent other unknown quantities in terms of x. There are no other unknown quantities to find, so we can skip this step.

16. Example: continued
Step 3 Write an equation in x that models the conditions. The camera’s original price minus the 40% reduction is the reduced price, $276. x − 0.4x = 276 Step 4 Solve the equation and answer the question. 0.6x = 276
x = 460
The camera’s price before the reduction was $460.

17. Example: continued
Step 5 Check the proposed solution in the original wording of the problem. The price before the reduction, $460, minus the 40% reduction should equal the reduced price given in the original wording, $276: 460 − 40% of 460 = 460 − 0.4(460) = 460 − 184 = 276. This verifies that the digital camera’s price before the reduction was $460.

18. Example: Solving a Formula for a Variable
Solve the formula P = 2l + 2w for l. First, isolate 2l on the right by subtracting 2w from both sides. Then solve for l by dividing both sides by 2. P = 2l + 2w P − 2w = 2l + 2w − 2w P − 2w = 2l

19. Example: Solving a Formula for One of its Variables
The total price of an article
purchased on a monthly
deferred payment plan is
described by the following
formula:
T is the total price, D is the
down payment, p is the
monthly payment, and m is
the number of months one
pays.
Solve the formula for p.
T – D = D – D + pm
T – D = pm
m m
T – D = p m