Section 3.6

More Applications of Linear Systems

3.6 Lecture Guide: More Applications of Linear Systems

Objective 1: Use systems of linear equations to solve word problems.

Strategy for Solving Word Problems

Step 1. Read the problem carefully to determine what you are being asked to find.

Step 2. Select a _________ to represent each unknown quantity. Specify precisely what each variable represents and note any restrictions on each variable.

Step 3. If necessary, make a _________and translate the problem into a word equation or a system of word equations. Then translate each word equation into an ____________ equation.

Step 4. Solve the equation or the system of equations, and answer the question completely in the form of a sentence.

Step 5. Check the _______________of your answer.

1. The bill for a cellular phone in August was $10 more than twice the September bill. The total that was required to pay both of these bills was $175. What was the bill for each month?

(a) Identify the variables:Let x = the cost of the phone bill for August

Let y = _____________________________________________________________________

1. The bill for a cellular phone in August was $10 more than twice the September bill. The total that was required to pay both of these bills was $175. What was the bill for each month?

(b) Write the word equations: (1) August bill is ______ more than __________ the September bill

(2) The total of the ______________ bill and the ________________ bill is ____________ .

1. The bill for a cellular phone in August was $10 more than twice the September bill. The total that was required to pay both of these bills was $175. What was the bill for each month?

(c) Translate the word equations into algebraic equations: (1) __________________________________

(2) __________________________________

1. The bill for a cellular phone in August was $10 more than twice the September bill. The total that was required to pay both of these bills was $175. What was the bill for each month?

(d) Solve this system of equations:

1. The bill for a cellular phone in August was $10 more than twice the September bill. The total that was required to pay both of these bills was $175. What was the bill for each month?

(e) Write a sentence that answers the question:

(f) Is this answer reasonable?

2. A hobbyist is making a picture frame out of wood molding. He has 42 inches of molding to create the frame. If the width of the frame must be 5 inches less than the length, what will be the dimensions of the frame? Will a 9 inch by 12 inch picture fit in this frame?

(a) Identify the variables (including the units of measurement):

Let L = the __________________ of the picture frame in inches

Let W = the _________________ of the picture frame in inches

2. A hobbyist is making a picture frame out of wood molding. He has 42 inches of molding to create the frame. If the width of the frame must be 5 inches less than the length, what will be the dimensions of the frame? Will a 9 inch by 12 inch picture fit in this frame?

Write the word equations:

(1) The _______________ of the frame is _________ inches.

(2) The _____________ is 5 inches less than the _________________ .

(b)

2. A hobbyist is making a picture frame out of wood molding. He has 42 inches of molding to create the frame. If the width of the frame must be 5 inches less than the length, what will be the dimensions of the frame? Will a 9 inch by 12 inch picture fit in this frame?

(c)Translate the word equations into algebraic equations:

(1) 2L + ___________ = ___________

(2) W = ________________

2. A hobbyist is making a picture frame out of wood molding. He has 42 inches of molding to create the frame. If the width of the frame must be 5 inches less than the length, what will be the dimensions of the frame? Will a 9 inch by 12 inch picture fit in this frame?

(d) Solve this system of equations:

2. A hobbyist is making a picture frame out of wood molding. He has 42 inches of molding to create the frame. If the width of the frame must be 5 inches less than the length, what will be the dimensions of the frame? Will a 9 inch by 12 inch picture fit in this frame?

(e) Write a sentence that answers the question:

(f) Is this answer reasonable?

Rate Principle

Applications of the Rate Principle

1. Variable cost = Cost per item Number of items2. Interest = Principal invested Rate Time3. Distance = Rate Time4. Amount of active ingredient = Rate of

concentration Amount of mixture5. Work = Rate Time

Amount = Rate×Base A R B

Mixture Principle for Two Ingredients

Amount in first + Amount in second = Amount in mixture

Applications of the Mixture Principle

1. Amount of product A + Amount of product B = Total amount of mixture

2. Variable cost + Fixed cost = Total cost

3. Interest on bonds + Interest on CDs = Total interest

4. Distance by first plane + Distance by second plane = Total distance

5. Antifreeze in first solution + Antifreeze in second solution = Total amount of antifreeze

Solve each of the remaining problems using the word problem strategy illustrated in the first two problems.

3. A small t-shirt screening business operates on a daily fixed cost plus a variable cost that depends on the number of shirt screened in one day. The total cost for screening 260 shirts on Friday was $1015. The total cost for screening 380 shirts on Saturday was $1345. What is the fixed daily cost? What is the cost to screen each shirt?

4. The Candy Shop has two popular kinds of candy. The owner is trying to make a mixture of 100 pounds of these candies to sell at $3 per pound. If the gummy gums are priced at $2.50 per pound and the sweet treats are $3.75 per pound, how many pounds of each must be mixed in order to produce the desired amount?

5. Ashley invested money in two different accounts. One investment was at 10% simple interest and the other was at 12% simple interest. The amount invested at 10% was $1500.00 more than the amount invested at 12%. The total interest earned was $480.00. How much money did Ashley invest in each account?

6. A hospital needs 80 liters of a 12% solution of disinfectant. This solution is to be prepared from a 33% solution and a 5% solution. How many liters of each should be mixed to obtain this 12% solution?

7. Two brothers decided to get together one weekend for a visit. They live 472 miles apart. They both left their homes at 8:00 a.m. on Saturday and drove toward each other. The younger brother drove 6 mi/h faster than the older brother and they met in 4 hours. How fast was each brother driving?