1. Math Club Week 2 Chapter 3 Linear Equations By John Cao

2. Officers Nominations for: Vice-President Secretary/Treasurer Historian Voting will take place at the next meeting.

3. Review of Chapter 1 What is 2 5 x 2 2 ? Answer: 2 7 : You add the exponents. What is (3 5 ) -2 ? Answer: 3 -10 What is (8) 2/3 ? Answer: 4 X 5/3 is 243. Find x. Answer: 27 What is √2/(√6 -2) Answer: √3 + √2 What is log 16 8 ? Answer: 3/4

4. KAHOOT TIME!

5. Chapter 3 Linear Equations 3.1 What is a Linear Equation? In the form y=mx+b 3.2 One equation, One variable With one equation you can only solve for one variable Solve 3x+5=11+x Answer: x=3 I hope that this was not a difficult problem.

6. Chapter 3 Linear Equations Solve the equation ax+b=c for x, where a, b, and c are constants. (x=?) Answer: x= (c-b)/a 3.3: Two Equations, Two variables These are systems of equations. 2 methods: Substitution, or Elimination Solve the system: 2x+4y=-1, and 3x-4y=7 Answer: (1, -1)

7. Chapter 3 Linear Equations Another problem: 2(√x) + 4(√y) =10 and 2(√x) -3(√y) = 3 Answer: (9,1)

8. Chapter 3 Linear Equations Example 3-12: Jim drives to his mom’s house, which is 40 miles away, and then drives back. On the way there he drives 40 mph, but on the way back he drives only 20 mph. What is his average speed? Hint it is not 30 mph. Distance=rate x time It will take him 1 hour to get there, and 2 hours to go back. The answer is 80/3= 26 and 2/3 mph.

9. Chapter 3 Linear Equations Example 3-13: A frog swims 8 miles downstream in 2 hours. She returns upstream in 14 hours. How fast does the frog swim in still water? Let the frog’s speed by x and the current be y. The frog’s rate downstream will be x+y, and the rate upstream is x-y. The frog is going 4 mph downstream, so you can have (x+y)(2)=8 or x+y=4 The frog is going 8/14 mph upstream, so you can have (x-y)(14)=8 or x-y=8/14 Doing systems, you have that the frog’s rate in still water is 16/7 mph.

10. Chapter 3 Linear Equations Example 3-14: Pipe A can fill a pool in 5 hours, while Pipe B can fill it in 4.How long will it take if they are operating at the same time? In 1 hour, pipe A can fill 1/5 of the pool. Pipe B can fill ¼ of the pool in 1 hour. That means that they can fill up ¼+1/5 of the pool in 1 hour. X(1/4 +1/5)=1, so x=20/9

11. Chapter 3 Linear Equations Example 3-15: Tom and Huck paint a fence for 4 hours, after which Jim helps them and they finish 2 hours later. If Jim had not helped them, it would have taken them 5 more hours to paint the fence. How long would it take for Jim to paint the fence alone? We first have 9(1/T + 1/H) =1 or (1/T +1/H) =1/9 Then we have 4(1/T +1/H) + 2(1/T + 1/H + 1/J)=1 Since (1/T + 1/H) = 1/9, we have 4/9 +2(1/9 + 1/J) =1 Solving for J we have that J=6, so he can paint the fence alone in 6 hours.

12. Chapter 3 Linear Equations Exercise 3.6- A canoeist paddled upstream for 2 hours, then downstream for 3. The rate of the current was 2 mph. When she stopped, the canoeist realized that she was 20 miles downstream from her starting point. How many hours will it take her to paddle back to her starting point? The paddler’s rate will be x. When going upstream, the paddler travels 2(x-2) miles upstream.

13. Chapter 3 Linear Equations She travels 3(x+2) miles downstream. Thus, by combining the equations, we have that she has paddled x+10 miles downstream. Since we know that she has travelled 20 miles, that means that x=10. Since she has to travel upstream, her rate will be 8 mph. She has to travel 20 miles, so 20/8 is 2.5.

14. Practice Problems 1. There are 16 coins in a bank. If the coins are all nickels and dimes and they total $1.05, how many nickels are there? Answer: 11 nickels 2. George Washington was born 11 years before Thomas Jefferson. In 1770 George Washington’s age was 3 more than 7 times the age of Jefferson in 1748. What was the sum of the two men’s ages in 1750? Answer: Washington’s age in 1748 is J+11. That means that his age in 1770 was J+11+12. That means that J+33=7J=3, so J=5. We have found out that Jefferson was 5 in 1748, so he was 7 in 1750 and Washington was 18. The sum of their ages was 25.

15. Practice Problems 3. Solve for c in terms of a and b given that √a+(b/c) = a√(b/c) Answer: First, we get rid of the square roots by squaring both sides, getting a+ (b/c) =(a 2 b)/c. Then you multiply both sides by c

16. Practice Problems 4. A train, x meters long, travelling at a constant speed, takes 20 seconds from the time it first enters a tunnel 300 meters long until the time it completely emerges from the tunnel. One of the stationary ceiling lights in the tunnel is directly above the train for 10 seconds. Find x. Answer: The train travels x meters in the 10 seconds which it is under the light. Thus its speed is x/10 meters per seconds (mps). In 20 seconds, the train travels x+300 m, since he has to be fully out of the tunnel. That means the rate is (300+x)/20m. Set the 2 equal to each other, and you would get x=300.

17. Practice Problems 5. Each valve A,B,C, when open, releases water into a tank at its own constant rate. With all 3 valves open, the tank fills in 1 hour, with only valves A and C open it takes 1.5 hour, and with only valves B and C open it takes 2 hours. How long will it take to fill the tank with only valves A and B open? Answer: 1/a + 1/b + 1/c = 1/1 1/a + 1/c = 1/1.5 1+b + 1/c = ½ 1/a + 1/b =1/x Solve, and you will find that x=1.2

18. UIL Introduction Number Sense: Number Sense: 80 mental math questions in 10 minutes. If you are really quick at calculations, this is for you. The problems get more difficult as you go on. Calculator: 60 Calculator problems in 30 minutes, 30 are straight typing, 12 are word problems, and 8 are “memorizing formula questions”. If you have fast fingers and are great at memorizing formulas, this is the one for you! Math: This is just straightforward hard math. It is 60 questions with a 40 minute time limit. Pre-Calculus is recommended, since there is some trig on it.

19. Next week will be our last meeting before school ends. We will focus entirely on UIL next week. Next Week: