1.3 Modeling with Linear Equations

In other words…..

Word Problems

Solving Real Life Problems!

• Step 1: Write a Verbal Model– Ex. Distance = Rate x Time

• Step 2: Create Labels– Ex. Distance = 20 miles

Rate = r Time = .2 hours

• Step 3: Write and Solve Algebraic Model– Ex. 20 = .2r

• Step 4: ANSWER THE QUESTION!!!

Example 1

You have accepted a job for which your annual salary will be $32,300. This salary includes year end bonus of $500. You will be paid twice in one month. What will your gross pay (pay before taxes) be for each paycheck?

Example 1 Continued

• Verbal ModelIncome for year = 24 Paychecks + Bonus

• LabelsIncome for year = $32,300Amount of each paycheck = xBonus = $500

• Algebraic Model32,300 = 24x + 500

Example 2

You have accepted a job that pays $8 an hour. You are told that after a two month probationary period, your hourly will be increased to $9 an hour. What percent raise will you receive after the two month period?

Example 2 Continued

• Verbal ModelRaise = Percent x Old wage

• LabelsOld wage = $8New wage = $9Raise = $9 - $8 = $1Percent = x

• Algebraic Model1 = x(8)

Example 3

Your family has an annual income of $57,000 and the following monthly expenses: mortgage ($1100), car payment ($375), food ($295), utilities ($240) and credit cards ($220). The total value of the monthly expenses represent what percent of your family’s annual income?

Example 3 Continued

• Verbal ModelMonthly Expenses = Percent x IncomeFor One Year

• LabelsIncome = 57,000Monthly Expenses = 26,760Percent = x

• Algebraic Model26,760 = x(57,000)

Example 4

• A rectangular kitchen is twice as long as it is wide, and its perimeter is 84 feet. Find the dimensions of the kitchen. (Find the length and the width)

Example 4 Continued

• Verbal Model2(Length) + 2(Width) = Perimeter

• LabelsPerimeter = 84 feetWidth = wLength = 2w

• Algebraic Model2(2w) + 2(w) = 84

Example 5

A plane is flying nonstop from Atlanta to Portland, a distance of about 2700 miles. After 1.5 hours in the air, the plane flies over Kansas City (a distance of 820 miles from Atlanta). Estimate the time it will take the plane to fly from Atlanta to Portland.

Example 5

• Verbal ModelDistance = Rate x Time

• LabelsDistance = 2700Time = tRate = distance to Kansas City =

time to Kansas City• Algebraic Model

2700 =

820

1.5820

1.5t

Example 6

• To determine the height of the Aon Center Building in Chicago, you measure the shadow cast by the building and find it to be 142 feet long. Then you measure the shadow cast by a four foot post and find it to be 6 inches long. Estimate the buildings height.

Example 6 Continued

• Verbal ModelHeight of building = Height of Post

Length of Building’s Length of Post’sShadow Shadow

• LabelsHeight of building = xBuildings shadow = 142 feetHeight of post = 4 feet = 48 inchesPost’s Shadow = 6 inches

• Algebraic Model 48

142 6

x

Homework

• Pg 107 #47-49,51-54, 57,62,63