Modeling with Linear Equations

Introduction to Problem Solving

Ex 1 Using a Verbal Model

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

Ex. 1

• Your income ($32,300) will be the 24 paychecks plus your $500 bonus.

• The unknown is the amount of each paycheck. Let’s make that x.

Ex. 1• Your income ($32,300) will be the 24 paychecks plus

your $500 bonus.• The unknown is the amount of each paycheck. Let’s

make that x.• Your pay can be expressed by the equation: 32,300 = 24x + 500

Ex. 1• Your income ($32,300) will be the 24 paychecks plus

your $500 bonus.• The unknown is the amount of each paycheck. Let’s

make that x.• Your pay can be expressed by the equation: 32,300 = 24x + 500 31,800 = 24x $1,325 = x

Ex. 2 Finding the Percent of a Raise

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

Ex. 2

• What do we know?• Original amount = 8• New amount = 9• Amount of change = 1

Ex. 2

• What do we know?• Original amount = 8• New amount = 9• Amount of change = 1• The equation we will use is:

amount of change = % change original amount

Ex. 2

• What do we know?• Original amount = 8• New amount = 9• Amount of change = 1• The equation we will use is:

amount of change = % change original amount

change%8

1 change%125. %5.12

Ex. 3 Finding the Percent of Monthly Expenses

• Your family has an annual income of $57,000 and the total monthly expenses of $26,760. Monthly expenses represents what percent of your family’s annual income?

Ex. 3

• Your family has an annual income of $57,000 and the total monthly expenses of $26,760. Monthly expenses represents what percent of your family’s annual income?

• The equation we will use here is: part = % whole

Ex. 3

• Your family has an annual income of $57,000 and the total monthly expenses of $26,760. Monthly expenses represents what percent of your family’s annual income?

• The equation we will use here is: part = % 26,760 = .469 or 46.9% whole 57,000

Ex. 4 Finding the Dimensions of a Room

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

Ex. 4 Finding the Dimensions of a Room

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

• We always want to make x the thing that we are comparing other things to. In this problem, we are comparing the length to the width.

Ex. 4

• This means that the width is x.• The length is twice the width, so that means that

Length = 2(width) or 2x.

Ex. 4

• This means that the width is x.• The length is twice the width, so that means that

Length = 2(width) or 2x.• The perimeter of a rectangle is: 2(length) + 2(width) = P

Ex. 4

• This means that the width is x.• The length is twice the width, so that means that

Length = 2(width) or 2x.• The perimeter of a rectangle is: 2(length) + 2(width) = P 2(2x) + 2x = P 6x=84, x = 14, so: w = 14ft, and l = 28ft

Ex. 5 A Distance Problem

• A plane is flying nonstop from Atlanta to Portland, a distance of about 2700 miles. After 1.5 hrs. 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.

Ex. 5• The equations we will use are: Rate x Time = Distance

or Distance = Time rate

or Distance = rate time

Ex. 5

• We know that it took 1.5 hrs. to fly 820 miles, so to find the rate, we use:

Distance = rate time

hm /66.5465.1

820

Ex. 5

• We now know the rate that the plane is flying. To find the time we use the equation:

distance = time rate

.94.466.546

2700hrs

Ex. 6 Similar Triangles

• To determine the height of the Aon Center Building, you must use similar triangles. We know that a 4 ft. post will cast a shadow of 6 in. The Building will cast a shadow of 142 ft. Set up a proportion to solve for the height of the building.

Ex. 6

• Remember, in similar triangles, the corresponding sides are in proportion.

Height of building = Shadow of Building Height of Post Shadow of Post

Ex. 6

• Remember, in similar triangles, the corresponding sides are in proportion.

Height of building = Shadow of Building Height of Post Shadow of Post• All units must be the same!!

5.

142

4x

ftx 1136

Literal Equations

• An equation that contains more than one variable is called a Literal Equation. Formulas are a great example of literal equations. Look at the formulas on page 103. Many you know, some you may not.

Literal Equations

• Here are some common equations that you will use in many of your math classes.

Literal Equations

• Here are some common equations that you will use in many of your math classes.

Ex. 9 Using a Formula

• A cylindrical can has a volume of 200 cubic centimeters and a radius of 4 cm. Find the height of the can.

Ex. 9

• A cylindrical can has a volume of 200 cubic centimeters and a radius of 4 cm. Find the height of the can.

• The formula for the volume of a can is:

hrV 2

Ex. 9

• Since we are trying to find the height, we will solve the equation for height first.

2r

Vh

hrV 2

Ex. 9

• Since we are trying to find the height, we will solve the equation for height first.

2r

Vh

hrV 2

2)4(

200

h .98.3 cmh

Percent Problems

• There is another type of percent problem that is very basic. Some examples are:

• What is 30% of 70?• 12 is 20% of what number?• 112 is what % of 300?

Percent Problems

• There is another type of percent problem that is very basic. Some examples are:

• What is 30% of 70?• 12 is 20% of what number?• 112 is what % of 300?• The word what becomes the variable x.• The word is now means equals.• The word of now means multiplication.

Percent Problems

• There is another type of percent problem that is very basic. Some examples are:

• What is 30% of 70?• 12 is 20% of what number?• 112 is what % of 300?• The word what becomes the variable x.• The word is now means equals.• The word of now means multiplication.

)70()30(. x

)()20(.12 x

)300()(112 x

Percent Problems

• There is another type of percent problem that is very basic. Some examples are:

• What is 30% of 70?• 12 is 20% of what number?• 112 is what % of 300?

21

)70()30(.

x

x

6020.

12

)()20(.12

x

x

x

%3737.300

112

)300()(112

orx

x

x

Class work

• Pages 105-106

• 12-22 even

Homework

• Pages 105-106

• 11-21 odd

• 35-53 odd

• 63