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3.6, Day #1 Systems of Linear Equations with Three Variables Slide 2 Do Now : The Troll Toll 1) Grab some slides. 2) Solve this brainteaser: You are on your way to visit your Grandma, who lives at the end of the valley. It's her birthday, and you want to give her the cakes you've made. Between your house and her house, you have to cross 7 bridges, but there is a troll under every bridge! Each troll, quite rightly, insists that you pay a troll toll. Before you can cross their bridge, you have to give them half of the cakes you are carrying, but as they are kind trolls, they each give you back a single cake. How many cakes do you have to leave home with to make sure that you arrive at Grandma's with exactly 2 cakes? Slide 3 Do Now (Periods 2 and 10/11) G-g-g-g-graded Do Now Take some slides and the half-sheet that says 3.5 Exit Card. We did not have time to do the Exit Card yesterday, so it will be a graded Do Now instead. Please put your stuff away and work on that. You will have 5 minutes. Slide 4 Clarification Standard form of plane ax+by+cz=d a,b,c NOT all zero This is why we could divide by d when we did the problem with the intercepts and the equation of the plane. -Exit Card- Be sure to answer the questions that are asked please. Write a function of x and y means your answer should include f(x,y)= Slide 5 Homework Concerns Were there any? Slide 6 Thats Space Ice Cream! Slide 7 Weigh the Wangdoodles http://www.mathplayground.com/wangdo odles.html Slide 8 Systems of Equations in Three Variables Slide 9 Slide 10 Solve this System Slide 11 Solution This is most easily done with substitution! For instance, solve the last equation for z, z=3-2x+4y. Now substitute that into the first equation. 2x+(3-2x+4y)=11 3+4y=11 y=2 Now we can plug that into the second equation to get x. X+2(2)=7, x=3. And similarly, we can now find z using the first equation. 2(3)+z=11, z=5 So the solution is (3,2,5) CHECK IT! Slide 12 Lets Do It! Slide 13 Lets Use Elimination! Goal: System of 2 equations in 2 variables Take any 2 of the equations, and eliminate a variable3x+2y+4z=11 (2x-y+3z=4)*24x-2y+6z=8 Lets eliminate y.7x+10z=19 Slide 14 The first two equations gave us 7x+10z=19. Now we can take the first and the third or the second and the third and get another equation with just xs and zs via elimination. I will take the second and the third. (2x-y+3z=4)*-3-6x+3y-9z=-12 5x-3y+5z=-1 -x-4z=-13 Now I have: Slide 15 Finishing it Up! Solve to see x=-3 and z=4. Plug these back into any equation to get y=2. The solution is then (-3,2,4). CHECK IT IN ALL 3 EQUATIONS! Slide 16 You Try! Slide 17 What Do You Notice? Slide 18 Multiply the top equation by -3 and combine with the middle equation. We end up with 0=8. NONSENSE! No solution! Slide 19 Combine the first and the second and get 2x+2y=4 Combine the second and the third and get 3x+3y=6 These are both just multiples of x+y=2. If we combined them, we would get 0=0. Infinitely many solutions! Slide 20 Application A public swimming pool has the following rates: ages under 5 are free, ages 5-16 are $3, and ages 16 and up are $4. The pool also has a policy that every child under age 5 must be accompanied by an adult. The families in your neighborhood decide to go to the pool as part of a summer party. There are 22 people in your group and an equal number of children under age 5 as people 16 years old and older. The total admission cost was $54. How many of each group went? X = ages under 5 Y = ages 5 -16 Z = ages 16 and up Slide 21 So there are 22 people in the group. This means: x+y+z=22 The total cost was $54 with the prices for each age group given to give: 3y+4z=54 Finally, there are an equal number of children under 5 as people 16 and over so that: x=z Slide 22 x=6, y=10, z=6 So 6 people under 5, 10 people 5-16 and 6 people 16 and up Slide 23 You Try! Courtney has a total of 256 points on three Algebra tests. His score on the first test exceeds his score on the second by 6 points. His total score before taking the third test was 164 points. What were Courtney s test scores on the three tests? Slide 24 Answer First=85, second=79, third=92 Slide 25 Chapter 3 Survey Slide 26 Other Games http://www.mathplayground.com/algebr aic_reasoning.html http://www.mathplayground.com/algebr aic_reasoning.html http://www.mathplayground.com/algebr a_puzzle.html http://www.mathplayground.com/algebr a_puzzle.html