section 3.6 more applications of linear systems. 3.6 lecture guide: more applications of linear...
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Section 3.6
More Applications of Linear Systems
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3.6 Lecture Guide: More Applications of Linear Systems
Objective 1: Use systems of linear equations to solve word problems.
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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.
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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 = _____________________________________________________________________
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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 ____________ .
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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) __________________________________
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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:
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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?
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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
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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)
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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 = ________________
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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:
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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?
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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
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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
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Solve each of the remaining problems using the word problem strategy illustrated in the first two problems.
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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?
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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?
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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?
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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?
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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?