1. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 2.4 - 1

2. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 2.4 - 2 Linear Equations and Applications Chapter 2

3. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 2.4 - 3 2.4 Further Applications of Linear Equations

4. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 4 2.4 Further Applications of Linear Equations Objectives 1.Solve problems about different denominations of money. 2.Solve problems about uniform motion. 3.Solve problems about angles.

5. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 5 2.4 Further Applications of Linear Equations Problems About Different Denominations of Money Problem-Solving Hint In problems involving money, use the fact that For example, 67 nickels have a monetary value of $.05(67) = $3.35. Forty-two five dollar bills have a value of $5(42) = $210. number of monetary units of the same kind Xdenomination= total monetary value.

6. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 6 2.4 Further Applications of Linear Equations Problems About Different Denominations of Money Elise has been saving dimes and quarters in a toy bank. Every time she saves a coin, she pulls a small lever, and the bank records the number of coins that have been deposited as well as the total amount in the bank. $29.95 202 coins The bank contains $29.95, consisting of 202 coins. How many of each type coin does the bank contain?

7. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 7 2.4 Further Applications of Linear Equations Problems About Different Denominations of Money Continued. Step 1 $29.95 202 coins Read the problem. The problem asks that we find the number of dimes and quarters that have been saved in the bank.

8. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 8 2.4 Further Applications of Linear Equations Problems About Different Denominations of Money Continued. Step 2 Assign a variable. Let x represent the number of dimes; then 202 – x represents the number of quarters. DenominationNumber of CoinsTotal Value $0.10x0.10x $0.25 202 – x 0.25(202 – x ) 202$29.95

9. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 9 2.4 Further Applications of Linear Equations Problems About Different Denominations of Money Continued. Step 3 Write an equation. DenominationNumber of CoinsTotal Value $0.10x0.10x $0.25202 – x0.25(202 – x) 202$29.95 Totals From the last column of the table,

10. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 10 2.4 Further Applications of Linear Equations Problems About Different Denominations of Money Continued. Step 4 Solve. Multiply by 100. Distributive prop. Subtract 5,050. Divide by –15.

11. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 11 2.4 Further Applications of Linear Equations Problems About Different Denominations of Money Step 5 Elise has 137 dimes and 202 – x = 202 – 137 = 65 quarters in the bank. Step 6 The bank has 137 + 65 = 202 coins, and the value of the coins is $.10(137) + $.25(65) = $29.95. State the answer. Check. Caution Always be sure your answer is reasonable! Continued.

12. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 12 2.4 Further Applications of Linear Equations Solving a Motion Problem (Opposite Directions) Two snowmobiles leave the same place, one going east and one going west. The eastbound snowmobile averages 24 mph, and the westbound snowmobile averages 32 mph. How long will it take them to be 245 miles apart? WE

13. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 13 2.4 Further Applications of Linear Equations Solving a Motion Problem (Opposite Directions) Step 1 Read the problem. We must find the time it takes for the two snowmobiles to be 245 miles apart. Caution The sum of their distances must be 245 mi. Each does not travel 245 mi. Continued.

14. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 14 2.4 Further Applications of Linear Equations Solving a Motion Problem (Opposite Directions) Step 2 Assign a variable. The sketch shows what is happening in the problem. Let x represent the time traveled by each snowmobile. 32 mph 24 mph Starting point Total distance = 245 mph RateTimeDistance Eastbound 24x24x Westbound 32x32x 245 Continued.

15. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 15 2.4 Further Applications of Linear Equations Solving a Motion Problem (Opposite Directions) Step 3Write an equation. RateTimeDistance Eastbound 24x24x Westbound 32x32x 245 Continued.

16. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 16 2.4 Further Applications of Linear Equations Solving a Motion Problem (Opposite Directions) Step 4Solve. Combine like terms. Divide by 56; lowest terms Continued.

17. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 17 2.4 Further Applications of Linear Equations Solving a Motion Problem (Opposite Directions) Step 4State the answer. The snowmobiles travel hr, or 4 hr and 22½ min. Step 5Check. Distance traveled by eastbound Distance traveled by westbound Total distance traveled OK Continued.

18. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 18 2.4 Further Applications of Linear Equations Solving a Motion Problem (Same Direction) Brandon works 360 miles away from his home and returns on weekends. For the trip home, he travels 6 hours on interstate highways and 1 hour on two-lane roads. If he drives 25 mph faster on the interstate highways than he does on the two-lane roads, determine how fast he travels on each part of the trip home.

19. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 19 2.4 Further Applications of Linear Equations Solving a Motion Problem (Same Direction) Step 2 Read the problem. We are asked to find the speed Brandon drives on the interstate highways and the speed he drives on the two-lane roads. Continued.

20. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 20 2.4 Further Applications of Linear Equations Solving a Motion Problem (Same Direction) Step 3 Assign a variable. The problem asks for two speeds. We can let Brandon’s speed on the two-lane highways be x. Then the speed on the interstate highways must be x + 25. For the interstate highways, and for the two-lane roads. Continued.

21. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 21 2.4 Further Applications of Linear Equations Solving a Motion Problem (Same Direction) Step 2Assign a variable. Summarizing this information in a table, we have: RateTimeDistance Interstate x + 25 6 6( x + 25) Two-lane x 1 x 360  total Continued.

22. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 22 2.4 Further Applications of Linear Equations Solving a Motion Problem (Same Direction) Step 3 Write an equation. RateTimeDistance Interstate x + 25 6 6( x + 25) Two-lane x 1 x 360 Continued.

23. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 23 2.4 Further Applications of Linear Equations Solving a Motion Problem (Same Direction) Step 3 Solve. Distributive prop. Collect like terms. Subtract 150. Inverse prop. Divide by 7. Continued.

24. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 24 2.4 Further Applications of Linear Equations Solving a Motion Problem (Same Directions) Step 5State the answer. Brandon drives the two-lane roads at a speed of 30 mph; he drives the interstate highways at x + 25 = 30 + 25 = 55 mph. Step 6Check by finding the distances using: Distance traveled on interstate highways Distance traveled on two-lane roads Total distance traveled 360 mi OK Continued.

25. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 25 2.4 Further Applications of Linear Equations Solving Problems Involving Angles of a Triangle From Euclidean Geometry The sum of the angle measures of any triangle equal 180º. AB C D E F

26. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 26 2.4 Further Applications of Linear Equations Finding Angle Measures Find the value of x and determine the measure of each angle in the figure.

27. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 27 2.4 Further Applications of Linear Equations Finding Angle Measures Step 1 Read the problem. We are asked to find the measure of each angle in the triangle. Step 2 Assign a variable. Let x represent the measure of one angle. Continued.

28. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 28 2.4 Further Applications of Linear Equations Finding Angle Measures Step 3 Write an equation. The sum of the three measures shown in the figure must equal 180º. Continued.

29. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 29 2.4 Further Applications of Linear Equations Finding Angle Measures Step 4 Solve the equation. Collect like terms. Subtract 110. Divide by 2. Continued.

30. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.4 - 30 2.4 Further Applications of Linear Equations Finding Angle Measures Step 5 State the answer. One angle measures 35º, another measures 2 x + 25 = 2(35) + 25 = 95º, and the third angle measures 85 – x = 85 – 35 = 50º. Step 6Check. Since 35º + 95º + 50º = 180º, the answer is correct. Continued.