PROBLEM SOLVING DISCIPLINE

Transcription

1 PROBLEM SOLVING DISCIPLINE YELLOW BELT TRAINING EQUATIONS NAME

2 Yellow Belt Order of Mastery 1. What is X? 2. Solving equations add/sub 3. Solving equations mult/div 4. Two-step equations 5. Variables on both sides 6. Multi-step equations 7. Translating and solving Okuden (secret teachings) 8. Payment Plans 9. Absolute Value Equations Name

3 What is X? So why is this called the Problem Solving Discipline? It s all about this mysterious letter known mostly as X. Algebra is just one big puzzle, a mystery you need to solve. Most of the time that thing you re solving for is referred to as X. It can be any letter really, or even an empty box. Remember seeing problems like this in elementary school? + 3 = 8 That empty box means you need to put a number in there to make that problem work. Most people can look at that and say, Well it s simply 5. 5 plus 3 equals 8. In Algebra we don t use that empty box anymore. We use a letter like X and it s called a VARIABLE. We call it that since the answer can vary in different problems and vary means change. It s the unknown part of the puzzle we need to figure out. So we d write it like this: x + 3 = 8 And in this case, X does indeed equal 5. But we don t have to use the letter X. We could use Y or Z or any letter for that matter. z + 3 = 8 It s the same problem, with a different letter used as the VARIABLE. This still says What number plus three equals eight? In this case Z does indeed equal 5. So we say Z = 5. It s very important to be able to translate what s going on in Algebra. Just like if you were learning another language like Spanish, you need to translate words back in forth. For instance if I said: What number minus three equals four? How would you write that as an EQUATION with a VARIABLE? Since we don t know the number, we ll call that X: x 3 = 4 Now what if I said write this phrase as an equation: Four is three less than what number? It actually says the same thing, but in a much more confusing way: x 3 = 4 That s why we need a translator! So let s make our own English to Algebra Dictionary.

4 What number? A number Unknown Variable ENGLISH ALGEBRA EXAMPLE Any letter (x, y, z, etc) A number plus 3 x + 3 Equals Is = Plus Added to Sum Increased by More than Minus Decreased by Subtracted by Difference Less than Fewer than Multiplied by Product of Times Of + ( ) or We don t use x anymore like we did in elementary school. It s already taken. A number plus 3 is 8 x + 3 = 8 A number increased by 6 x + 6 The difference between 5 and a number 5 x The product of 5 and 3 (5)(3) or 5 3 The product of 5 and a number 5x Divided by Quotient Is less than (note how similar to subtract) Is greater than (note how similar to add) or / < > The quotient of 8 and a number 8 x or 8/x A number is less than three x < 3 Three is greater than a number 3 > x

5 Now that you have your universal translator you re ready to try some phrases on your own: Write the phrase as an equation with a variable: 1. A number decreased by 3 8. Difference of 10 and a number 2. The sum of 5 and a number is less than the difference of 25 and a number. 3. Product of 4 and a number 10. A number divided by 4 is increased by a number 11. The product of 2 and a number is less than a number 12. A number increased by 2 is 4 6. The sum of 20 and a number is A number divided by 22 is less than 3 7. A number increased by 10 is greater than times a number is 56. or equal to 44. Take the equation and describe it in a sentence with words like above: 15. 9n 16. n + 5 = x = < 21 x 19. 3a > x = 18

6 A World in Balance Solving Equations with Addition and Subtraction You re going to be asked to solve the equation a lot over the next few years. But what exactly does that mean? It simply means find out what number the letter equals. In a more precise and boring way to put it: Find a value for the variable that makes the equation true. We have several tricks to help you with this. But first you must know that all equations must have BALANCE. That s why we have the picture of the scale. I think you would all agree with the following statement: 4 = 4 There s a value of 4 on the left of the equal sign, and a value of 4 on the right. They are balanced. If I add 3 to the left side, they are no longer balanced: But if I add three to both sides, it s balanced again: = 4 No!!! = Yes!!! That seems pretty obvious. But it comes in handy when solving an equation. All I have to remember is this: ANYTHING YOU DO TO ONE SIDE, YOU HAVE TO DO TO THE OTHER Consider the following equation: x 3 = 10 You might be able to just look at that and know that x is 13. But I m going to prove it to you. I want x by itself on one side. If I did that, then it would say X = some number. So how do I do that? x 3 = 10 I need to get rid of that 3 part. So I will add the opposite. (remember Since the problem is: the inverse property?) What s the opposite of 3? It s simply + 3. So let s add 3 to the left side. But remember, whatever I do to the left side, I must do to the right side. So I need to add 3 to EACH side. x 3 = x = 13 Since = 0, I m left with x on the left side, and = 13 on the right. So that s your answer, x = 13 But to make sure, you should always CHECK YOUR WORK If you say x = 13, then I should be able to plug in 13 for x and the equation will be true (BALANCED). (13) 3 = = 10 I M RIGHT!!!!!

7 Now you try! Solve each equation for the missing variable = r /3 + x = 2/ = 2 + y 12. 2/5 = a 1/5 3. n 5 = r + ¾ = ¼ 4. a 3/5 = x = t 3/8 = (-x + 3) = x + 7 = (5 x) + 13 = m - (-13) = t (-4) = x = = a + (-4) = m w t - (-16) = g Challenge: You decide to try to ride the elevator to street level (floor 0) without pushing any buttons. The elevator takes you up 4 floors, down 6 floors, up 1 floor, down 8 floors, down 3 floors, up 1 floor and then down 6 floors to street level. Write and solve an equation to find what floor you started on.

8 Review Solve these first Keeping the Balance Solving Equations by Multiplying and Dividing 1. x 12 = b + 7 = ½ + x = -4/ /4 + c = 2/ m = (-9) r = 20 Solve the following equations by using division. To get the variable by itself, remember the creed: ANYTHING YOU DO TO ONE SIDE, YOU HAVE TO DO TO THE OTHER So to get rid of that 3 in front of the r in problem 7, you need to divide by 3..on both sides. 7. 3r = n = y = x = x = x = x = b = = -2a Woohooo!!! Fractions!!! I love fractions!!!! Solve by multiplying 16. x t = = d = m = 4 43 h = h = -16

9 Wheww!!! I thought the fractions were done..now I have more!!!! Alright!!! Solve using the reciprocal k = y = x = m = x = c = Real life problems 28. It takes 45 peanuts to make one ounce of peanut butter. How many peanuts will be needed to make a 12 ounce jar of peanut butter? Write the equation and solve it. 29. You ate 3 of the 8 slices of a pizza. You paid $3.30 as your share of the total cost of the pizza. How much did the whole pizza cost? Write the equation and solve it. 30. A homeowner is installing a fence around his garden as shown. If the perimeter is 220 feet, what are the dimensions of each side? Challenge: A hot dog vendor at a soccer game sells one half of his supply of hot dogs during the first half of the game. During intermission, he sells a total of five hot dogs. During the second half, he sells three eighths of his original supply. He is left with only four hot dogs. How many did he have to start with?

10 Follow the Steps! Just two for now Ok so we ve solved by adding and subtracting, then solved by multiplying and dividing. But what happens if we put them together? No problem! Just do one step at a time. The first step is to add or subtract. Example: 3x 5 = 10 First I add 5 to both sides to get rid of that 5. 3x 5 = x = 15 Now ISOLATE the variable by dividing 3x = and we get x = Now you try! Solve the equation for each variable 1. 2x + 7 = p 16 = = 3y n 3n 4 = x 12x = c (-7c) 8 2 = 18 OOOhhhhh!! Parentheses. Remember to distribute. 7. 5(6 + j) = (k 2) = (4 m) = x + 4(x + 3) = y (8 + 6y) = x 2(3x 2) = -6

11 Woooohoooo! More fractions! More Fractions! (x + 9) = (x + 3) = Real life 15. Your school s drama club charges $4 per person for admission to a play. The club borrowed $400 from parents to pay for costumes and props. After paying back the parents, the drama club has $100. How many people attended the play? Choose the equation that represents this situation, and then solve it. a. 4x = 100 b. 4x = 400 c. 4x 400 = 100 d. 4x 100 = The bill for the repair of your car is $458. The cost of the parts is $339. The cost of labor is $34 per hour. Which equation could you use to find the number of hours of labor? a. 34(x + 339) = 458 b. 34x = 458 c x = 458 d x = You have a 90 pound calf you re raising to enter into the Del Mar Fair. You expect it to gain 65 pounds per month. In how many months will the animal weigh 1000 pounds? Write the equation and solve it (2y + 1) 6y = x + 2(3x + 5) = x + (5x 7) = a (2a 1) = -2 Challenge: Jason went out to buy some computer games. He spent half of what he had plus $5 at the first store. At the second store, he spent half of what was left plus $4, and at the third store, he spent half of the remainder plus $3. He then had $5 left at the end of the day. How much money did he start with?

12 Wheel of Fortune Variables on both sides!!! Noooooo!!!!! Up until now, we ve been dealing with a variable on just one side. But it s just not always that easy. Sometimes we have variables on both sides. If that s the case, we need to take care of that first. Before we add or subtract or multiply or divide we need to get the VARIABLES ON THE SAME SIDE. Example: n + 1 = -13 6n To get the variable on the same side, we do what we ve always done, use the inverse property. I ve got an n on the left, and I ve got a 6n on the right. Let s take the opposite of the 6n by adding 6n. And remember: ANYTHING YOU DO TO ONE SIDE, YOU HAVE TO DO TO THE OTHER So let s add 6n to BOTH sides: n + 1 = -13 6n +6n +6n Now I m right back where I was the night before with just two steps left: 7n + 1 = -13 First subtract, then divide: n = -14 So x = Each space has a number on it. If you get an answer that has that number, place the letter of that answer above that number. The goal is to find out what the puzzle says while secretly learning how to solve linear equations. Not every answer will belong in the puzzle, so you need to find out the value of each letter. Show your work for every problem to get credit. 1) E = 5E ) - 5D - 8 = - 6D ) 3 + 3S = 7S ) 2(2N + 3) = -6(N + 9)

13 5) 6C + 9 = - 3C + 9 6) - 4A + 1 = 2 5A 7) 3 + 2R = 7R - 7 8) 2L - 7 = L 9) 1 + B = - B 13 10) 6M + 4 = M 11) 8i (8 + 6i) = 20 12) - 4x + 10 = 65 + x 13) 5 2W = 2W ) -3(4V + 1) + 6V = 4(2V 6) 15) 6(O + 3) + 3O = 3(O 2) 16) - 5(6H + 9) = ) 1 18) -2(4F + 2) = -2(F + 3) + 9 (3G 12) = 6 2(G 1) 3

14 THIS IS HOW YOU DO IT!!!!!! Ok so now you ve got some mad skills, so let s put them all together. It s going to get ugly. EXAMPLE: 2(5x -12) = 2x 8 + 4x Follow these 3 steps to solve EVERY equation: STEP 1: SIMPLIFY EACH SIDE That means distribute and combine on EACH side first before you start moving anything around. So we distribute the 2 on the left side: 2(5x 12) = 2x 8 + 4x 10x 24 = 2x 8 + 4x Then combine like terms on the right side: 10x 24 = 2x 8 + 4x 10x 24 = 6x 8 STEP 2: VARIABLES ON THE SAME SIDE That means you need to have X only on one side. It could be the left or the right. Just get rid of one of them by adding its opposite. 10x 24 = 6x 8-6x -6x 4x 24 = -8 STEP 3: ISOLATE THAT VARIABLE First you add or subtract: Then you multiply or divide: 4x 24 = -8 4x = x = 16 x = You should always check your answer by plugging it back into the original equation to see if it s right (wherever there s an x, put a 4): 2(5 4 12) = (20 12) = (8) = = 16 It checks out!

15 Now you try! Solve the following equations for the missing variable: 1) 20 = 4x 6x 2) 6 = 1 2n + 5 3) 8x 2 = 9 + 7x 4) a + 5 = 5a + 5 5) 4m 4 = 4m 6) p 1 = 5 p + 3 p 8 7) 5p 14 = 8 p + 4 8) p 4 = 9 + p 9) 8 = (x + 4) 10) 12 = 4( 6x 3) 11) 14 = ( p 8) 12) (7 4x) = 9 13) 18 6k = 6(1 + 3k) 14) 5n + 34 = 2(1 7n) 15) 2(4x 3) 8 = 4 + 2x 16) 3n 5 = 8(6 + 5n) 17) (1 + 7x) 6( 7 x) = 36 18) 3(4x + 3) + 4(6x + 1) = 43 19) 24a 22 = 4(1 6a) 20) 5(1 5x) + 5( 8x 2) = 4x 8x

16 Riddle Me This! You have some problem solving skills and now you need to put them to good use. Use your knowledge of translations and equations to figure out the following riddles. 1) a. Three hundred less than fifteen times Brian's age is six less than his age. How old is Brian? Jose? b. One hundred forty-seven less than three times Jose's age is twenty-seven less than his age. How old is c) If Jasmine were three times as old as she was three years ago, she will be forty-five less than six times her current age. How old is Jasmine? d) If Sydney were three times as old as she was three years ago, she will be forty-one less than five times her current age. How old is Sydney? 2) a. One-seventh of a number, decreased by 5 is 8. What is the number? b. The difference between 393 and half of a number is 110. What is the number? c. Eighteen exceeds one-ninth of a number by 15. What is the number? d. Twenty exceeds one-half of a number by 11. What is the number?

17 3) a. Forty-eight less than Kevin's age is one hundred fifty less than three times his age. How old is Kevin? b. Ninety-three less than three times Caleb's age is thirteen less than his age. How old is Caleb? c. If Courtney were three times as old as she was three years ago, she will be fifty-five less than five times her current age. How old is Courtney? d. If Natalie were five times as old as she was six years ago, she will be ninety-eight more than her current age. How old is Natalie? Now to make sure you ve truly fine tuned your equation solving skills, here s some more practice. These are about as ugly as you ll ever see. 4. (x 5) = 2x 5x x + 1 = 4x / / 4 x = x + 1 / x + 18 = (-5x 10) x + 4 = x x - 9 / 10 = 3x + 21 / x = -7x x 11. (x 5) + 2x = 15 3x

18 A little review first to show your extreme intelligence: Payment Plans (a real-life example of why we need Algebra) 1. -2(x + 5) = -6x (4x + 3) 5 = -8x (x + 2) = 4 1 (12x + 4) 5x 4. 2(8 4x) = 3 1 (33 18x) + 3 Now to those real-life scenarios 5. Blockbuster charges $8 to rent the Halo2 video game for five days. Membership to Blockbuster is free. Nate s video game club charges only $3 to rent a game for five days, but membership in the club is $50 per year. Compare the costs of the two rental plans. If you planned on renting the game once a month over the course of a year, which plan is the better deal? Write an equation to prove your results. 6. A rock climbing gym charges nonmembers $16 per day to use the gym and $8 per day for equipment rental. If you become a member, you pay a yearly fee of $450 for unlimited climbing and $6 per day for equipment rental. Compare the two plans. Write an equation and solve it to find how many times you must use the gym to justify becoming a member. 7. You want to paint a piece of pottery. The total price is the cost of the piece plus an hourly painting rate. At Kamil s Craft Shack, you can buy a vase for $12 and they charge $7 an hour to paint. Alli s Awesome Art Depot sells a similar vase for $15 and lets you paint for $4 an hour. If it takes you 2 hours to paint a vase, which place should you go to? At what rate would it be the same to go to either place? 8. Mr. T. decides to charge his students to use the laptops each day. If you re not a member of Mr. T s new program, it ll cost you $5 a session each time you use the laptop. But if you become a member, with a low-low initiation fee of $70, it ll cost you only $3 a session to use the laptop. How many times would you have to use the laptop to justify joining Mr. T s lame membership?

19 To be negative, or not to be negative That s the absolute value We know that the absolute value of a number can t be negative, because it really represents a distance. It s the distance a number is from zero on a number line. --distance-- --distance I I I So when we say, what s the absolute value of -4: -4 = 4 But what if we asked: Solve the equation x = 8 There are really two answers. If x = 8, then 8 = 8 If x = -8, then -8 = 8 The equation has two solutions: 8 and -8 So the answer to that equation is: 8, -8 Since the absolute value of a number is never negative, solving this equation is impossible: x = -10 has NO SOLUTION Solve the following equations 1. x = x = x + 1 = 3 4. x 2 = x = x = 3 7. x + 6 = x = x - 2 = x 2 = x 4 = x + 5 = x - 3 = x = x = 3 Write an absolute value equation that has the given solutions and and 10 Challenge: The distance between the earth and the sun changes because the earth s orbit around the sun is an ellipse. The maximum distance from earth to the sun is 94.5 million miles. The minimum distance is 91.4 million miles. Write an absolute value equation that has the minimum and maximum distances between the earth and sun as its solutions.

20 ANSWERS TO PROBLEM SOLVING YELLOW BELT EQUATIONS WHAT IS X? 1. x x 3. 4x x 5. x x = x + 10 > x < 25 x 10. x/4 = x = x + 2 = x/22 < x = The product of 9 and a number 16. The sum of a number and 5 is The product of 2 and a number is is less than the diff. of 21 and a number 19. The product of 3 and a number is greater than The difference of 28 and a number is 18 A WORLD IN BALANCE / / / / / /4 20. ½ CHALLENGE: X = 0 So 17 th floor KEEPING THE BALANCE / / / / / X 22. 4/ X= X X=$ x + 2x + 3x + 2x = 220 x = 22 66ft by 44ft 8 Challenge: 72 FOLLOW THE STEPS 1) 4 7) 3 13) 11 18) 7 2) 14 8) 8 14) 6 19) 2 3) 2 9) 9 15) C 20) 1/2 4) 5 10) 1 16) B 21) -1 5) 6 11) 14 17) 65x + 90 = 1000 Challenge: $90 6) 2 12) 2 x = 14 WHEEL OF FORTUNE 1. E. = 8 2. D = 6 3. S = N = C = 0 6. A = 1 7. R = 2 8. L = 5 9. B = M = I = X = W = V = 3/2 15. O = H = G = F = - 7/6 GIRAFFES HAVE NO VOCAL CHORDS THIS IS HOW YOU DO IT 1) 2 2) 0 3) 7 4) 0 5) No solution. 6) 1 7) 6 8) No solution. 9) 4 10) 0 11) 6 12) 4 13) 1 14) 4 15) 3 16) 1 17) 5 18) 4 19) No solution. 20) 5

21 RIDDLE ME THIS 1. a) 15b 300 = b 6 b=21 b) 3j 147 = j 27 j=60 c) 3(j 3) = 6j 45 j=12 d) 3(s 3) = 5s 41 s=16 2. a) 1/7x 5 = 8 x=91 b) 393 1/2x = 110 x=566 c) 1/9x + 15 = 18 x = 27 d) 1/2x + 11 = 20 x=18 3. a) k 48 = 3k 150 k=99 b) 3c 93 = c 13 c=40 c) 3(c 3) = 5c 55 c=23 d) 5(n 6) = n + 98 n= / /2 PAYMENT PLANS x = 3x + 50 Nate s cheaper if rent more than ten x + 8x = x 25 equal 26 or more to justify x = x Alli s cheaper at 2 hours same at 1 hour x = 3x times same 36 or more to justify TO BE NEGATIVE OR NOT THAT S THE ABSOLUTE VALUE , no solution 2. no solution 16. I x 13 I = , I x 2 I = , I x 9 I = , , no solution 8. 18, / , , 1/ /3, /2, , -7