Lesson 1: Introduction to Equations

Transcription

1 Solving Equations Lesson 1: Introduction to Equations Solving Equations Mentally Key to Solving Equations

2 Solving Equations Lesson 2: Solving Addition Equations Golden Rule for Solving Equations: In order to solve addition equations, we will use the opposite operation: Example 1 Example 2

3 Solving Equations Lesson 2: Solving One-Step Addition Equations Directions: Solve each equation on your own paper. If your answer is a decimal, round to the nearest hundredth. 1. x + (-7) = y + 8 = 2 3. x + (-2) = 1 4. r +.10 = y + (-12) = p + (-21) = 2 7. q = x + 12 = w + (-10) = y = = h = y + ½ 13. ¾ = r + ½ p = y = q = 20 Bonus: 17. Jerry went clothes shopping. 2 shirts and 3 pairs of shorts cost $ After tax his total was $ How much was the sales tax? (Solve the equation for t) Cost of clothes + tax = price (This is called a verbal model) t = You are hiking a 15 mile trail. You estimate that you ve already hiked 6.5 miles. How many more miles do you need to hike in order to finish the trail? (Write an equation and solve) # of miles hiked + # of miles not hiked = total miles

4 Solving Equations Directions: Solve each equation. Show every step. (2 points each) 1. b + (-18) = = h y + 1/3 = 5/ p = h + (-19) = 22

5 Solving Equations Lesson 2: Solving One-Step Addition Equations Answer Key Directions: Solve each equation. If your answer is a decimal, round to the nearest hundredth. 1. x + (-7) = y + 8 = 2 x + (-7) (-7) = 10 (-7) y +8-8 = 2-8 x = 17 y = x + (-2) = 1 4. r +.10 = -8 x + (-2) (-2) = 1 (-2) r = x = 3 r = y + (-12) = p + (-21) = 2 y (-12) = -27 (-12) p + (-21) (-21) = 2 (-21) y = -15 p = q = x + 12 = 42 q = x = q = x = w + (-10) = y = -5.2 w + (-10) (-10) = 18 (-10) y = w = 28 y = -7

6 Solving Equations = h = y + ½ h = 22 y + ½ = 9 h = y + ½ - ½ = 9 ½ h = 20.1 y = 8.5 or y = 8 & 1/2 13. ¾ = r + ½ p = -17 r + ½ = ¾ p = r + ½ - ½ = ¾ - ½ p = -26 r = ¼ y = q = y = q = y = 42 q = -3 Bonus: 17. Jerry went clothes shopping. 2 shirts and 3 pairs of shorts cost $ After tax his total was $ How much was the sales tax? (Solve the equation for t) Cost of clothes + tax = price (This is called a verbal model) t = t = t = 3.18 (Jerry paid $3.18 in sales tax) 18. You are hiking a 15 mile trail. You estimate that you ve already hiked 6.5 miles. How many more miles do you need to hike in order to finish the trail? (Write an equation and solve) # of miles hiked + # of miles not hiked = total miles m = m = m = 8.5 (You need to hike 8.5 more miles in order to finish the trail,)

7 Solving Equations Directions: Solve each equation. Show every step. (2 points each) 1. b + (-18) = -13 b+ (-18) (-18) = -13 (-18) Subtract -18 from both sides b = : -13 (-18) = b = 5 : = = h = h Subtract 42 from both sides -10 = h : = 32+(-42) = y + 1/3 = 5/6 y+1/3-1/3 = 5/6 1/3 Subtract 1/3 from both sides y = 1/2 : 5/6 1/3 = 3/6 = 1/ p = p = Subtract 6.5 from both sides p = : = (-6.5 ) = h + (-19) = 22 h +(-19) (-19) = 22 (-19) Subtract -19 from both sides h = 41 : 22 (-19) = = 41

8 Lesson 3: Solving Subtraction Equations In order to solve subtraction equations, we will use the opposite operation: Example 1 Example 2

9 Lesson 3: Solving One-Step Subtraction Equations Directions: Solve each equation on your own paper. If your answer is a decimal, round to the nearest hundredth. 1. r - (-10) = s (-8) = x 1.8 = y 22 = y - (-9) = w 25.4 = x (-38) = t (-13) = x (-1.8) = y 55 = = p (-5) = h = y y (-18) = t ¾ = 16/4 16. h 2/3 = 5/6 Bonus: 17. Judy bought a computer and printer set $ Judy used a coupon for $15 towards her purchase. What was the original price of the computer set? (Solve the equation for c) Original Price coupon = Total price C 15 = Five months ago, Lisa started a rigorous exercise program and diet. She lost 32.5 pounds on the program and she now weighs lbs. How much did Lisa weigh originally? Original Weight Weight loss = New Weight

10 Directions: Solve each equation. (2 points each) 1. s (-13) = t ½ = -3/4 3. r 6.9 = = h p - (-33) = - 9

11 Lesson 3: Solving One-Step Subtraction Equations Answer Key 1. r - (-10) = s (-8) = -15 r (-10) + (-10) = 21 + (-10) s (-8) + (-8) = (-8) r = 11 s = x 1.8 = y 22 = -10 x = y = x = -.4 y = y - (-9) = w 25.4 = y (-9) + (-9) = 12 + (-9) w = y = 3 w = x (-38) = t (-13) = 25 x (-38) + (-38) = -9 + (-38) t (-13) + (-13) = 25 + (-13) x = -47 t = x (-1.8) = y 55 = 14 x (-1.8) + (-1.8) = (-1.8) y = x = -4.7 y = 69

12 = p (-5) = h 8 p (-5) = -28 h 8 = 23 p (-5) + (-5) = -28 +(-5) h = p = -33 h = = y y (-18) = 38 y 10 = -8 y (-18) + (-18) = 38 + (-18) y = y = 20 y = t ¾ = 16/4 16. h 2/3 = 5/6 t ¾ +3/4 = 16/4 + ¾ h 2/3 + 2/3 = 5/6 + 2/3 t = 19/4 h = 3/2 Bonus: 17. Judy bought a computer and printer set $ Judy used a coupon for $15 towards her purchase. What was the original price of the computer set? (Solve the equation for c) Original Price coupon = Total price C 15 = C = C = (The original price of the computer sets was $563.38) 18. Five months ago, Lisa started a rigorous exercise program and diet. She lost 32.5 pounds on the program and she now weighs lbs. How much did Lisa weigh originally? Original Weight Weight loss = New Weight W = W = W= 176 (Lisa originally weighed 176 lbs)

13 Directions: Solve each equation. (2 points each) 1. s (-13) = 12 s (-13) +(-13) = 12 +(-13) Add -13 to both sides s = -1 : 12+(-13) = t ½ = -3/4 t ½ + ½ = -3/4 + ½ Add ½ to both sides t = -1/4 : -3/4 + ½ = -1/4 3. r 6.9 = 9.2 r = Add 6.9 to both sides r = 16.1 : = = h = h Add 27 to both sides -36 = h : = p - (-33) = - 9 p (-33)+(-33) = -9 +(-33) Add -33 to both sides P = -42 : -9 + (-33) = -42

14 Lesson 4: Solving Multiplication Equations In order to solve multiplication equations, we will use the opposite operation: Example 1 Example 2 A Few Notes About Fractions: Example 3 Example 4

15 Lesson 4: Solving Multiplication Equations Directions: Solve each equation on your own paper. If your answer is a decimal, round to the nearest hundredth x = x = x = x = x = x = /3x = /7x = -6/ /9x = y = ¾ = -2/3x 12. 1/2x = = -3j k = p = /7 = 3/7a BONUS: people registered for a volleyball tournament. How many 6 person teams can be formed? Number of people on a team Number of teams = Total number of registrants 6 t = 140 6t = 140 (Solve for t) 18. John makes $12.45 an hour at his part-time job. Last week he made $ How many hours did John work? Pay rate Number of hours worked = Total Pay check

16 Directions: Solve each equation. Show every step. (2 points each) 1. -6r = /3x = x = /4y = -3/ = -9y

17 Lesson 4: Solving Multiplication Equations Answer Key Directions: Solve each equation. If your answer is a decimal, round to the nearest hundredth x = x = 36 10x = x = x = 8 x = x = x = 17 9x = x = x = 9 x = -17/2 or x = x = x = x = x = 2.4 x = /3x = /7x = -6/7 =109 = x = x = -3/2 or /9x = y = 8 =16 x = -18.8y = y = 10

18 11. ¾ = -2/3x 12. 1/2x = 5.5-2/3x = ¾ (2) 1/2x = 5.5(2) (-3/2) -2/3x = ¾(-3/2) x = 11 x = -9/ = -3j k = -45-3j = k = j = 19/3 k = -45/ p = /7 = 3/7a 9.3p = 82.1 (7/3) 3/7a = -6/7 (7/3) a = -2 p = 8.83 BONUS: people registered for a volleyball tournament. How many 6 person teams can be formed? Number of people on a team Number of teams = Total number of registrants 6 t = 140 6t = 140 (Solve for t) 6t = t = 23.3 (23 6 person teams can be formed. 18. John makes $12.45 an hour at his part-time job. Last week he made $ How many hours did John work? Pay rate Number of hours worked = Total Pay check h = h = h = 14 (John worked 14 hours)

19 1. -6r = 84-6r/-6 = 84/-6 Divide by -6 on both sides r = -14 : 84/-6 = /3x = -14 (-3/2)-2/3x = -14(-3/2) Multiply both sides by (-3/2) (the reciprocal of 2/3 x = 21 : -14(-3/2) = x = 8.4.7x/.7 = 8.4/.7 Divide both sides by.7 x = 12 : 8.4/ /4y = -3/5 (4/3)3/4y = -3/5(4/3) Multiply by the reciprocal: 4/3 on both sides y = -8/10 : (-3/5)(4/3) = -8/ = -9y -135/-9 = 9y/-9 Divide both sides by = y : -135/-9 = 15

20 Lesson 5: Solving Division Equations In order to solve division equations, we will use the opposite operation: Example 1 Example 2

21 Lesson 5: Solving Division Equations Directions: Solve each equation on your own paper. If your answer is a decimal, round to the nearest hundredth. 1. = 9 2. =18 3. = = = =. 7. = = 9. = 10. = = = BONUS: 13. You would like to give each of your 5 friends $12 each for helping you clean your yard. How much money do you need? (Solve for t) Total Amount / Number of friends = Amount for each friend. t / 5 = 12 =12

22 1. = = 5 3. = 4. = 5. 16=

23 Lesson 5: Solving Division Equations Answer Key Directions: Solve each equation. If your answer is a decimal, round to the nearest hundredth. 1. = 9 2. =18 3 = =18 2 y = 27 x = = =4.5 5 = =4.5 8 r = -100 x = = = = = 2.2. p = = r 7. = = 5 = = -7 x = 27.5 p = -1.4 or -7/5 9. = 10. = = = = y y = 160

24 11. = = = = 14 t = = p BONUS: 13. You would like to give each of your 5 friends $12 each for helping you clean your yard. How much money do you need? (Solve for t) Total Amount / Number of friends = Amount for each friend. t / 5 = 12 =12 5 =12 5 t = 60 (You would need $60 in order to give your 5 friends $12 each)

25 1. = 12 4 = 12 4 Multiply by 4 on both sides y = -48 : = = 5 8 = 5 8 Multiply by -8 on both sides x = 40 : -5-8 = =. 5 = 5 Multiply by 5 on both sides r = -1/2 : -1/10(5) = -1/2 4. =. = Multiply by the reciprocal (3/2) 5/4 = y : (3/2 2/3 = 5/4) 5. 16=. 16 ( )=( ) Multiply both sides by -9/2 72 = r : -16(-9/2) = 72

26 Solving One-Step Equations Mixed Review Problems Directions: Solve each equation on your paper. If your answer is a decimal, round to the nearest hundredth x = y (-7) = r +(-8) = = = x +(-47) 6. 1/3x = p 28 = /5y = 8/10 9. = y + (-28) = = p = -6r = h k = /3j = = Directions: Solve each equation. Show every step. (2 points each) 1. -6x = y 14 = r +(-10) = = = x + 19

27 Solving One-Step Equations - Mixed Review Problems Answer Key Directions: Solve each equation. If your answer is a decimal, round to the nearest hundredth x = y (-7) = 26-3x = -44 y (-7) + (-7) = 26 + (-7) -3-3 y = 19 x = r +(-8) = r + (-8) (-8) = -12 (-8) = 9 = r = -4 t = = x + (-47) 6. ⅓x = -6.6 x + (-47) = 15 3 ⅓x = x + (-47) (-47) = 15 (-47) x = 62 x = p 28 = p = p = -27 y = -2 =. = 9. =. 10. y + (-28) = = y + (-28) (-28) = (-28) p = y = -16.8

28 = p = -6r p 6 = -27-6r = 90 p = p = -21 r = = h k = -24 h + 16 = 62 k + 5 = -24 h = k = h = 46 k = /3j = = (3/2) 2/3j = -24(3/2) ( 2) = 24 ( 2) j = -36 h = x = 108-6x/-6 = 108/-6 Divide by -6 on both sides x = -18 : 108/-6 = y 14 = - 45 y = Add 14 to both sides y = -31 :

29 3. r +(-10) = 82 r+(-10)- (-10) = 82 (-10) Subtract -10 r = 92 : 82-(-10) = = = 13 9 = 13 9 Multiply by -9 on both sides t = 117 : = = x = x Subtract 19 from both sides -14 = x : 5-19 = -14

30 Lesson 6: Two Step Equations Order for Solving Two-Step Equations Example 1 Example 2

31 Lesson 6: Two Step Equations - Continued Example 3 Example 4 Example 5

32 Lesson 6: Solving Two-Step Equations Directions: Complete each problem on your own paper. Show all steps! 1. 6x 18 = x + (-6) = x/4 +6 = /3x 4 = x/5 3 = /5 x +5 = x 2 = x + 10 = = 3/4x = y/ x +5x -4 = x 7 4 3x= x +3 2x 9 = /2x +3 +1/2 x = = 2x x = 3x x +2x 17. 3/4x /3x = = 4.4x x /5x + 3/10x 5 = = 7x 3x + 9 7x BONUS: 21. You earn extra money by babysitting. One person pays you $20 a week. Other people pay you $7 per hour. One week you made $55. How many hours did you babysit? (Solve for h) Amount charged per hour Number of hours + constant charge = Total earned 7 h + 20 = 55 7h + 20 = 55

33 Directions: Solve each equation. Show every step. (2 points each) 1. -7x + 7 = x x 2 = y/5 18 = x 8.2 9x = Jason has a lawn mowing business where he charges $3 for gas and $7 per hour. He earned $27.50 from one job. How many hours did he mow? 7x + 3 = 27.50

34 Lesson 6: Solving Two-Step Equations Answer Key 1. 6x 18 = x + (-6) = -27 6x = x + (-6) (-6) = -27 (-6) 6x = 51-3x = x = 17/2 or 8.5 x = = ⅔x 4 = = ⅔x 4+4 = = -18 ⅔x = 2 4 = x = 2 x = -72 x = = x +5 = = = -7 x +5-5 = x = = x = x = -35 x = 50

35 7. 8x 2 = x + 10 = 53 8x = x = x = 44 -x = x = 5.5 x = = 3/4x = y/ /4x 9 = 3 y/8 + 4 = 18 3/4x = y/ = /4x = 12 y/8 = 14 (4/3) 3/4x = 12 (4/3) (8) y/8 = 14(8) x = 16 y= x +5x -4 = x 7 4 3x= 10 8x 4 = -12 2x -3x -7-4 = 10 8x = x 11 = 10 8x = -8 -x = x = x = -1 x = x +3 2x 9 = ½x ½x = 10 5x 2x = -21 ½x + ½x +3 = 10 3x -6 = -21 x +3 = 10 3x 6+6 = x +3-3 =10-3 3x = -15 x = x = -5

36 = 2x x = 3x x +2x 2x x = 82 3x x + 2x = 54 2x 9x = 82 3x + 4x + 2x -6-3 = 54-7x 2 = 82 9x 9 = 54-7x = x = x = 84 9x = 63-7x/-7 = 84/-7 9x/9 = 63/9 x = -12 x = /4x /3x = = 4.4x x 1 3/4x + 2/3x + 3 = x x 1 = /12x + 3 = x + 6.8x = /12x = x -9.1 = /12x = x = x = (12/17) 17/12x = 25.5(12/17) x = x/11.2 = / 11.2 x = /5x + 3/10x 5 = = 7x 3x + 9 7x 7/10x 5 = x 3x + 9 7x = 36 7/10x = x 3x 7x + 9 = 36 7/10x = x + 9 = 36-3x = 36-9 (10/7) 7/10x = 17.5(10/7) -3x = 27 x = 25-3x/-3 = 27/-3 x = -9 BONUS: 21. You earn extra money by babysitting. One person pays you $20 a week. Other people pay you $7 per hour. One week you made $55. How many hours did you babysit? (Solve for h) Amount charged per hour Number of hours + constant charge = Total earned 7 h + 20 = 55 7h + 20 = 55 7h = h = 35 7h/7 = 35/7 h = 5 (You babysat 5 hours that week)

37 1. -7x + 7 = -91-7x = Subtract 7 from both sides -7x = -98 : = -98-7x/-7 = -98/-7 Divide by -7 on both sides x = 14 : -98/-7 = x x 2 = x+3x+8-2 = x +6 = x+6-6 = Rewrite like terms together Combine like terms Subtract 6 from both sides 9x = -138 : = x/9 = -138/9 Divide by 9 on both sides x = -46/3 : -138/9 = -46/3 3. y/5 18 = 7 y/ = 7+18 Add 18 to both sides y/5 = 25 : 7+18 = 25 y/5(5) = 25(5) Multiply by 5 on both sides y = 125 : 25(5) = 125

38 x 8.2 9x = x-9x = Rewrite like terms together x = Combine like terms x = Add 4.8 to both sides -2x = -13 : = -13-2x/-2 = -13/-2 Divide by -2 on both sides x = 6.5 : -13/-2= Jason has a lawn mowing business where he charges $3 for gas and $7 per hour. He earned $27.50 from one job. How many hours did he mow? 7x + 3 = x+3 = 27.5 Original Problem 7x +3-3 = Subtract 3 from both sides 7x = 24.5 : = x/7 = 24.5/7 Divide by 7 on both sides x = 3.5 : 24.5/7 = 3.5

39 Lesson 7: Equations with the Distributive Property Steps for Solving Equations with the Distributive Property Example 1

40 Example 2 Example 3

41 Lesson 7: Equations with the Distributive Property 1. 5( x-6) = (y 8) = (4 + x) = (y 14) = (7 y) = (x + 8) 4 = /2(x 12) + 8 = (x 1/2) + 4x = x + 3(x 4) +2 = x (4 x) + 3 = (x 1) + 3(2x +1 ) = = 2 (x+3) 5(x 1) 13. 3x 2(2x 4) = = (2x 1) +2x 15. 3(2x 1) 4(2x +2) = x + 2(x -5) -.5(2x +18) = 32 For problems 17-20, decide whether the given value is a solution to the equation. If not, find the correct solution (2x 1) (x + 4) -2 = t 2(4 t) = 17 x = -3 t = z + 3(6+2z) 7 = (w 2) 5 (3-3w) = -31 z = w = 0 Directions: Solve each equation. Show every step. (2 points each) 1. -4(x+ 5) = /3(y 3/2) = (w + 12) = = 2x + 5(-2x 7) 5. 2(b 6) 5(b + 3) = 15

42 Lesson 7: Solving Equations with the Distributive Property Answer Key 1. 5( x-6) = (y 8) = 16 5x 30 = 50-3y + 24 = 16 5x = y = x = 80-3y = x = 16 y = 8/3 3. (4 + x) = (y 14) = x = y +7 = x = y +7-7 = x = y = x = 6 y = (7 y) = (x + 8) 4 = y = 19 3x = -9-9y = x +20 = -9-9y = -44 3x = x = -29 y = 44/9 3 3 x = -29/3

43 7. -½(x 12) + 8 = (x 1/2) + 4x = 21 -½x = -22-6x +3 +4x = 21 -½ x+ 14 = -22-6x +4x +3 = 21 -½x = x +3 = 21 -½x = -36-2x +3-3 = ½x = x = 18 x = x = x + 3(x 4) +2 = x (4 x) + 3 = -35 4x + 3x = -18 2x 4 + x +3 = -35 7x 10 = -18 2x +x 4 +3 = -35 7x = x -1 = -35 7x = -8 3x 1 +1 = x = -34 x = -8/7 3 3 x = -34/ (x 1) + 3(2x +1 ) = = 2 (x+3) 5(x 1) 2x -2 +6x +3 = -7 2 (x+3) 5(x 1) = -28 2x +6x = -7 2x +6-5x +5 = -28 8x +1 = -7 2x -5x = -28 8x +1-1 = x +11 = -28 8x = -8-3x = x = x = -1 x = 13

44 13. 3x 2(2x 4) = = (2x 1) +2x 3x -4x +8 = (2x -1) +2x = 24 -x +8 = x 3 +2x = 24 -x +8-8 = x +2x 5-3 = 24 -x = 14 8x -8 = x 8 +8 = x = -14 8x = x = (2x 1) 4(2x +2) = x + 2(x -5) -.5(2x +18) = 32 6x -3 8x -8 = -18-3x +2x -10 x 9 = 32 6x 8x -3-8 = -18-3x +2x x 10-9 = 32-2x -11 = -18-2x -19 = 32-2x = x = x = x = x = 7/2 x = -51/2

45 For problems 17-20, decide whether the given value is a solution to the equation. If not, find the correct solution (2x 1) (x + 4) -2 = t 2(4 t) = 17 x = -3 t = 25 YES x = -3 is correct NO; t = 25 is not correct 3(2-3 -1) (-3 +4) 2 = -24 3t 8 + 2t = 17 3(-7) (1) 2 = -24 3t + 2t -8 = = -24 5t 8 = = -24 5t = t = 25 5t/5 = 25/5 t = z + 3(6+2z) 7 = (w 2) 5 (3-3w) = -31 z = w = 0 No; z = is not correct Yes; w = 0 is correct 8z z 7 = -87 8(0-2) 5(3-3 0) = -31 8z + 6z = -87 8(-2) 5(3) = z + 11 = = z = = z = z/14 = -98/14 z = -7

46 1. -4(x+ 5) = -52-4x -20 = -52 Distribute -4 throughout the parenthesis -4x = Add 20 to both sides -4x = -32 : = -32-4x/-4 = -32/-4 Divide by -4 x = 8 : -32/-4 = /3(y 3/2) = 6 1/3y ½ = 6 Distribute 1/3 throughout the parenthesis 1/3y ½ + ½ = 6 + ½ Add ½ to both sides 1/3y = 13/2 or 6.5 : 6+ ½ = 13/2 or 6.5 (3)1/3y = 6.5(3) Multiply by 3 on both sides Y = 19.5 : 6.5(3) = (w + 12) = w 36 = -45 Distribute -3 throughout the parenthesis w = -45 Rewrite like terms together -18 3w = -45 Combine like terms: = w = Add 18 to both sides -3w = -27 : = -27-3w/-3 = -27/-3 Divide by -3 on both sides w = 9 : -27/-3 = 9

47 4. 21 = 2x + 5(-2x 7) 21 = 2x -10x 35 Distribute the 5 throughout the parenthesis 21 = -8x 35 Combine like terms: 2x-10x = -8x = -8x Add 35 to both sides 56 = -8x : = 56 56/-8 = -8x/-8 Divide by -8 on both sides -7 = x : 56/-8 = (b 6) 5(b + 3) = 15 2b 12 5b 15 = 15 Distribute 2 throughout the first parenthesis & The -5 throughout the 2 nd parenthesis 2b 5b = 15-3b- 27 = 15 Rewrite like terms together Combine lie terms -3b = Add 27 to both sides -3b = 42 : = 42-3b/-3 = 42/-3 Divide by -3 on both sides b = -14 : 42/-3 = -14

48 Bonus Lesson: How to Study for Quizzes and Tests In order to study for a test or quiz, you should create a study guide. A study guide allows you to write down any important rules or definitions that you ll need to remember, and will allow you to practice problems that will be assessed on the quiz or test. The following directions will help you to create a study guide: 1. Find out what concepts/skills will be assessed, if possible. Make a list of the skills. 2. Create a Study Guide: Dedicate one page for each concept. On the page include: Any important definitions, rules or information. 4-5 practice problems of varying difficulty. Don t forget word problems, if applicable. (I usually write these on the back since they take up more room). 3. Practice Problems: Look through your notes and find 4-5 practice problems that you have the correct solution to. You must have the correct solution so that you can check your answer. **It s OK to do a problem more than once. Most likely, the problems that you choose, you completed at least a few days ago. Write the practice problem on your study guide. Make sure you know where to find the solution. TIP: Highlight the problems that you use for the study guide, so that you can go back and reference the solutions easily. 4. Once your study guide is set-up, complete all the problems that you wrote down. Check the answers with your solutions. If you did poorly on any particular concept, go back and repeat the process with different problems. Keep practicing until you are comfortable and achieving success. The following two pages are a template that can be used for the first quiz. I completed some parts of the template (in red) for you to help guide you through creating your first study guide. Use your notes to fill in the blanks and complete the problems.

49 Study Guide for Quiz #1 Concept 1: One-Step Equations 1. Rule: Whatever you do to one side of the equation, 2. Rule: Instead of dividing by a fraction, you must multiply by the. Pg 9 # 3 x + (-2) = 1 Addition Equations Subtraction Equations Pg 11 #2 s (-8) = -15 Pg 9 # 7 q = Pg 11 #6 w 25.4 = Multiplication Equations Pg 13 # 3-9x = -81 Pg 15 # 3 r/5 = -20 Division Equations Pg 13 # 7-2/3x = 109 Pg 15 # 5 p/1.9 = 7.7 Pg 13 # 9-8/9x = 16

50 Concept 2: Two-Step Equations Rule 1: Always remove the first by Rule 2: Then remove the coefficients using HINT: Complete the bottom by choosing at least 4 two-step problems from your notes. Write the problem, solve and then check your answer. Hopefully you are now ready for Quiz #1! Don t worry it s only 6 problems. Good Luck!

51 Equations Quiz #1 Directions: For questions 1 and 2, circle the answer that best fits the question. 1. Which of the following steps would you use to solve the equation: ⅔x = 6? a). Divide both sides of the equation by 6. b). Multiply both sides of the equation by ⅔. c). Multiply both sides of the equation by 3/2. d). Subtract ⅔ from both sides of the equation. 2. What is the correct value for x in the equation: 2x 3 = -13 a). x = -5 b). x = -8 c). x = 5 d). x = 8 Directions: Solve each equation. Show each step of solving the equation x = = x (-5) 3. ¼x +8 = y 10 = 6 2

52 Equations Quiz #1 Answer Key Directions: For questions 1 and 2, circle the answer that best fits the question. 1. Which of the following steps would you use to solve the equation: ⅔x = 6? a). Divide both sides of the equation by 6. b). Multiply both sides of the equation by ⅔. c). Multiply both sides of the equation by 3/2. d). Subtract ⅔ from both sides of the equation. 2. What is the correct value for x in the equation: 2x 3 = -13 a). x = -5 2x -3 = -13 b). x = -8 2x = c). x = 5 2x = d). x = 8 x = -5 Directions: Solve each equation. Show each step of solving the equation! 1. -3x = = x (-5) -3x = -33 x (-5) = x (-5) +(-5) = 12+(-5) x = 7 x = ¼x +8 = y 10 = 6 2 ¼x +8 = 9.5 y = ¼x +8-8 = (2)y = 16(2) 2 ¼ x = 1.5 4(¼ x) = 1.5(4) y = 32 x = 6

53 Lesson 8: Equations with Fractions Fractions are not easy numbers to deal with, so if your equation contains fractions, your first step will be to eliminate the fractions. Steps to eliminate the fraction: 1. If you have only 1 fraction (or 2 or more fractions with the same denominator), multiply ALL terms by the denominator. This will eliminate the fraction, and keep your equation balanced. 2. If you have 2 or more fractions (with different denominators), multiply ALL terms by the least common denominator (lcd). Example 1

54 Example 2 Example 3

55 Lesson 8: Solving Equations with Fractions 1. =10 2. = ⅓(x +4) = x + ½(x 3) = x ⅔(x 2) = 9 6. ¾ x 9 +¼(x -2) = = ⅜x ⅝(x 2) x + ⅓( x 3) = ¼(2x 4) = ¾ (3y -4) + ¼(2y - 2) = y - ½(2y -6) -3y = ½ 2 4 = /3(y 5) ¼(4 y) = j 5/9j = 4 For problems, 15 18, decide whether the given solution is correct. If not, find the correct solution /6x 3(x 5) = /5 2/5(y + 10) +1/10 = -1.7 x = 25.5 y = /3(p 8) + ½(p +2) = /5 2/3(y 7) = -1.4 P = 17 y = -4 Directions: Solve each equation. (2 points each) 1. = /3(x - 8) = x ½(x + 12) = ¾ x ¼(x - 9) = **5. ½(x-4) + ¾(x+ 5) = 13

56 Lesson 8: Solving Equations with Fractions Answer Key 1. = 2. =. 2 = = x +4 = 20 2x -2 = -36 x +4-4 = x = x = 16 2x = x = ⅓(x +4) = x + ½(x 3) = 15 3[⅓(x +4)] = -6(3) 2[5x + ½(x 3)] = 15(2) 1(x+4) = x +1(x -3) = 30 x +4 = x +x -3 = 30 x +4-4 = x -3 = 30 x = x = x = x = 3

57 5. 2x ⅔(x 2) = 9 6. ¾ x 9 +¼(x -2) = 23 3[2x ⅔(x 2)] = 9(3) 4[¾ x 9 +¼(x -2)] = 23(4) 6x -2(x -2) = 27 3x 36 +1(x -2) = 92 6x 2x +4 = 27 3x 36 +x -2 = 92 4x +4 = 27 3x +x 36 2 = 92 4x +4-4 = x -38 = 92 4x = 23 4x = x = 130 x = 23/4 4 4 x = = ⅜x ⅝(x 2) x + ⅓( x 3) = -10 ⅜x ⅝(x 2) +4 = -20 3[-2x + ⅓( x 3)] = -10(3) 8[⅜x ⅝(x 2) +4] = -20(8) -6x +1(x -3) = -30 3x 5(x-2) + 32 = x + x -3 = -30 3x 5x = x = x + 42 = x = -27-2x = x = -202 x = 27/ x = 101

58 9. ¼(2x 4) = ¾ (3y -4) + ¼(2y - 2) = 28 4[¼(2x 4)] = 18(4) 4[¾ (3y -4) + ¼(2y - 2)] = 28(4) 1(2x -4) = 72 3(3y -4) +1(2y -2) = 112 2x -4 = 72 9y y -2 = 112 2x = y +2y 12-2 = 112 2x = 76 11y 14 = y = x = 38 11y = y = 126/ y - ½(2y -6) -3y = ½ = 2[3y - ½(2y -6) -3y] = -21(2) 2[ + + ½ ]= 6y 1(2y -6) 6y = -42 x (2x -4) = 24 6y -2y +6-6y = -42 x +4 +2x -4 = 24 6y-2y -6y +6 = -42 x +2x +4-4 = 24-2y +6 = -42 3x +0 = 24-2y +6-6 = x = 24-2y = x = 8 y = 24

59 13. 2/3(y 5) ¼(4 y) = j 5/9j = 4 12[2/3(y-5) ¼(4-y) = 3(12) 9[j 5/9j] = 4(9) 8(y-5) 3(4-y) = 36 9j 5j = 36 8y y = 36 4j = 36 8y + 3y = 36 4j/4 = 36/4 11y -52 = 36 j = 9 11y = y = 88 11y /11 = 88/11 y = 8 For problems, 15 18, decide whether the given solution is correct. If not, find the correct solution /6x 3(x 5) = /5 2/5(y + 10) +1/10 = -1.7 x = 25.5 y = -5 NO: x = 25.5 is not correct. YES, y = -5 is correct 6[1/6x 3(x-5) = -36(6) 1/5 2/5(-5+10) +1/10 = -1.7 x 18(x-5) = (5)+.1 = -1.7 x 18x + 90 = = x = = x = x/-17 = -306/-17 x = /3(p 8) + ½(p +2) = /5 2/3(y 7) = -1.4 P = 17 y = -4 YES; p = 17 is correct NO, y = 10 is not correct. 1/3(17 8) + ½(17 + 2) = [3/5 2/3(y-7) = -1.4(15) 1/3(9) + ½(19) = (y-7) = = y + 70 = = y +79 = y = y = y /10 = -100/-10 y = 10

60 Directions: Solve each equation. Show every step. (2 points each) 1. = 18 3 = 18 3 Multiply by 3 on both sides 3x 2 = -54 : = -54 3x = Add 2 to both sides 3x = -52 : = -52 3x /3 = -52/3 x = -52/3 Divide by 3 on both sides (Leave you answer as a fraction) 2. 2/3(x - 8) = 6 3[2/3(x-8)] = 6(3) 2(x-8) = 18 Multiply by 3 on both sides to get rid of fraction 2x-16 = 18 Distribute 2 throughout the parenthesis 2x = Add 16 to both sides 2x = 34 : = 34 2x/2 = 34/2 Divide by 2 on both sides x = 17 : 34/2 = 17

61 3. -4x ½(x + 12) = [-4x ½(x+12)] = 23.25(2) Multiply by 2 to get rid of the fraction -8x (x+12) = x x 12 = 46.5 Distribute -1 throughout the parenthesis -9x 12 = 46.5 Combine like terms: -8x x = 9x -9x = Add 12 to both sides --9x = 58.5 : = x/-9 = 58.5/-9 Divide by -9 on both sides x = -6.5 Simpify: 58.5/-9 = ¾ x ¼(x - 9) = [3/4x+18-1/4(x-9)] = 21.75(4) 3x + 72 (x-9) = 87 3x + 72 x + 9 = 87 3x x +72+9= 87 Multiply by 4 to get rid of the fractions Distribute -1 throughout the parenthesis Rewrite like terms together 2x +81=87 Combine like terms; 3x-x = 2x & 72+9=81 2x = Subtract 81 from both sides 2x = 6 : 87-81=6 2x2 = 6/2 Divide by 2 on both sides x = 3 : 6/2=3

62 **5. ½(x-4) + ¾(x+ 5) = 13 4[1/2(x-4) + ¾(x+5)] = 13(4) 2(x-4) + 3(x+5) = 52 Multiply all terms by 4 to get rid of both fractions 2x-8 + 3x+15 = 52 Distribute the 2 and 3 throughout the parenthesis 2x + 3x = 52 5x + 7 = 52 5x+7-7 = 52-7 Rewrite like terms together Combine like terms Subtract 7 from both sides 5x = 45 : 52-7 = 45 5x/5 = 45/5 Divide by 5 on both sides x = 9 : 45/5 = 9

63 Lesson 9: Literal Equations Example 1 Example 2

64 Lesson 9: Solving Literal Equations 1. Solve this formula for b. 2. Solve this formula for p. A = ⅓b Solve this formula for n. 4. Solve this formula for k. = h = ¼k 9 5. The formula for finding the perimeter of a rectangle is: P=2L+2W. (2 length + 2 width) w L Solve this formula for W. If the perimeter of the rectangle is 64m and the length is 18m, find the width of the rectangle. Justify your answer mathematically.

65 6. The formula for finding the volume of a box is: V = lwh (length width height.) Solve this formula for h. The volume of the box is 540 in 2. The length is 15 in. and the width is 4 in. What is the height of the box? Justify your answer mathematically. 7. The formula for finding the area of a triangle is: A = ½bh (½ base height) Solve this formula for h. h b If the area of the triangle is 114 cm 2 and the base of the triangle is 19 cm, find the height of the triangle. Justify your answer mathematically.

66 8. The formula for finding the volume of a cylinder is: V=. (Pi radius (squared) height) Solve this equation for h. The volume of the cylinder is cm 3. The diameter is 8 cm. Find the height of the cylinder. (Use 3.14 for pi.) Explain how you found your answer. 9. Solve the equation for q. 10. Solve the equation for t. qr+2 = h p = 3t -2 p s 1. Solve for b. (1 point) 2. Solve for m. (1 point) ab 9 = 11 =

67 3. The formula for finding the perimeter of a rectangle is: P=2L+2W. (2 length + 2 width) (3 points) w L Solve this formula for L. If the perimeter of the rectangle is 132m and the width is 22m, find the length of the rectangle. Justify your answer mathematically.

68 Lesson 9: Solving Literal Equations Answer Key 1. Solve this formula for b. 2. Solve this formula for p. A = ⅓b +5 ⅓b +5 = A ⅓b +5-5 = A -5 ⅓b = A-5 5( ) =(r)5 p+q = 5r 3(⅓b) =( A-5) 3 p +q q = 5r -q b = 3(A-5) p = 5r - q b = 3A Solve this formula for n. 4. Solve this formula for k. = h = ¼k 9 2( )=j(2) ¼k 9 = h nm+1 = 2j ¼k 9 +9 = h +9 nm +1-1 = 2j -1 ¼k = h +9 nm = 2j -1 4(¼k) = (h +9)4 m m k = 4(h +9) n = 2j-1 m k = 4h +36

69 5. The formula for finding the perimeter of a rectangle is: P=2L+2W. (2 length + 2 width) w Solve this formula for W. P=2L+2W 2L +2W = P 2L -2L +2W = P -2L 2W = P 2L 2 2 W = P-2L 2 L If the perimeter of the rectangle is 64m and the length is 18m, find the width of the rectangle. Justify your answer mathematically. P = 64 m L = 18 m W =? W = P-2L Justify: P=2L+2W (Use original 2 equation) W = 64 m 2(18 m) 2 64 m= 2(18 m) +2(14 m) W= 28 m 2 64 m = 64 m W = 14 m

70 6. The formula for finding the volume of a box is: V = lwh (length width height.) Solve this formula for h. V = lwh V = lwh lw lw V = h lw The volume of the box is 540 in 2. The length is 15 in. and the width is 4 in. What is the height of the box? Justify your answer mathematically. V = 540 in 2 L = 15 in W = 4 in V = h Justify: lw V = lwh 540 in 3 = h (15 in 4 in) V = 15 in 4 in 9 in 9 in = h V = 540 in 3

71 7. The formula for finding the area of a triangle is: A = ½bh (½ base height) Solve this formula for h. A = ½bh ½bh = A 2(½bh) = A 2 Bh = 2A B B H = 2A B h b If the area of the triangle is 114 cm 2 and the base of the triangle is 19 cm, find the height of the triangle. Justify your answer mathematically. A = 114 cm 2 B = 19 cm H =? H = 2A B Justify: H = cm 2 A = ½bh 19cm 114 cm 2 = ½ 19cm 12cm H = 228 cm 2 19 cm 114 cm 2 = 114 cm 2 H = 12 cm

72 8. The formula for finding the volume of a cylinder is: V=. (Pi radius (squared) height) Solve this equation for h. V= h h = V h = V H = V The volume of the cylinder is cm 3. The diameter is 8 cm. Find the height of the cylinder. (Use 3.14 for pi.) Explain how you found your answer. V = cm 3 Diameter = 8 cm Radius = 4 cm (The radius is ½ of the diameter, so 8/2 = 4) H = V H = cm (4 2 ) Check: V= cm 3 = (3.14)(4 cm ) 2 (3 cm) H = cm cm 3 = cm (4 cm) 2 H = cm 3 H = 3 cm cm 2 I used the formula: H = V/. I substituted for the volume which was given in the problem. I knew the diameter was 8cm and the radius of a circle is ½ the diameter, so ½ of 8 is 4. I substituted 4 for the radius into the formula and solved for H. I checked my answer by substituting the values for the radius and height into the original equation, V = h. Since the volume equaled cm 3, I knew my answer was correct.

73 9. Solve the equation for q. 10. Solve the equation for t. qr+2 = h p = 3t -2 p s p (qr+2) = h(p) p 3t -2 = p s qr +2 = hp s (3t -2) = p(s) s qr +2-2 = hp -2 3t 2 = ps qr = hp -2 r r 3t 2 +2 = ps +2 q = hp -2 3t = ps +2 r 3t = ps t = ps Solve for b. (1 point) ab 9 = 11 ab 9+9 = 11+9 Add 9 to both sides ab = 20 : 11+9 = 20 ab/a = 20/a b = 20/a Divide by a on both sides

74 2. Solve for m. (1 point) + 4 = 4[ ]= (4) nm+p = 4q nm+p p = 4q-p nm = 4q-p nm/n = (4q-p)/n m = (4q-p)/n Multiply both sides by 4 to get rid of fraction Subtract p from both sides Divide by n on both sides 3. The formula for finding the perimeter of a rectangle is: P=2L+2W. (2 length + 2 width) (3 points) w L Solve this formula for L. 2l + 2w = p 2l + 2w 2w = p- 2w 2l = p-2w 2l /2 = (p-2w)/2 L = (p-2w)/2 Subtract 2w from both sides Divide by 2 on both sides If the perimeter of the rectangle is 132m and the width is 22m, find the length of the rectangle. Justify your answer mathematically. P = 132 m w = 22 m L = (p-2w)/2 L = (132-2(22))/2 L = 44 m Formula for finding L from above Substitute values for P and w Evaluate for L

75 Justify: Use the original formula and substitute the P = 2L + 2W values. If both sides are equal, then your 132 = 2(44) + 2(22) answer is correct. 132 = = 132

76 Lesson 10: Equations with Variables on Both Sides Goal Example 1

77 Lesson 10: Equations with Variables on Both Sides - Continued Example 2 Example 3

78 Steps to Solving Equations Equation: Step 1: Does your equation have fractions? YES Multiply EVERY term in the equation (on both sides) by the denominator or the LCD if you have 2 or more fractions. NO Go to step 2 Step 2: Does your equation involve the distributive property? (Do you see parenthesis with more than one term inside?) YES Rewrite the equation using the distributive property for the terms involved. NO Go to step 3. Step 3: On either side of the equation, do you have like terms that can be combined? YES Rewrite the equation with like terms together. Don t forget to take the sign in front of the term! Then rewrite the equation again, combining the like terms. NO Go to Step 4. Step 4: Do you have variables on both sides of the equation? YES Add or subtract the terms to get all the variables on one side and all the constants on the other side. Then go to Step 6. NO Go to Step 5. Step 5: At this point, you should have a basic two-step equation to solve. Use addition or subtraction to remove any constants from the variable side of the equation. Step 6: Use multiplication or division to remove any coefficients from the variable side of the equation. Step 7: Check your answer

79 Steps to Solving Equations Equation: Step 1: Does your equation have fractions? YES Multiply EVERY term in the equation (on both sides) by the denominator or the LCM if you have 2 or more fractions. NO Go to step 2 Step 2: Does your equation involve the distributive property? (Do you see parenthesis with more than one term inside?) YES Rewrite the equation using the distributive property for the terms involved. NO Go to step 3. Step 3: On either side of the equation, do you have like terms that can be combined? YES Rewrite the equation with like terms together. Don t forget to take the sign in front of the term! Then rewrite the equation again, combining the like terms. NO Go to Step 4. Step 4: Do you have variables on both sides of the equation? YES Add or subtract the terms to get all the variables on one side and all the constants on the other side. Then go to Step 6. NO Go to Step 5. Step 5: At this point, you should have a basic two-step equation to solve. Use addition or subtraction to remove any constants from the variable side of the equation. Step 6: Use multiplication or division to remove any coefficients from the variable side of the equation. Step 7: Check your answer

80 Steps to Solving Equations - Example Equations with Variables on Both Sides Use the 7 Solving Equations Steps to solve the following equation: ⅔(x+3) +2 = 3x +⅓ (3x -4) Step 1: Does your equation have fractions? YES Multiply EVERY term in the equation (on both sides) by the denominator. NO Go to step 2 Step 2: Does your equation involve the distributive property? (Do you see parenthesis with more than one term inside?) YES Rewrite the equation using the distributive property for the terms involved. NO Go to step 3. Step 3: On either side of the equation, do you have like terms that can be combined? YES Rewrite the equation with like terms together. Don t forget to take the sign in front of the term! Then rewrite the equation again, combining the like terms. YES: Remove the fraction by multiplying every term by 3 (the denominator). I highlighted each term a different color. Each term was multiplied by 3. ⅔(x+3) +2 = 3x +⅓ (3x -4) 2(x+3) + 6 = 9x + 1(3x-4) 2(x+3) + 6 = 9x + 1(3x-4) 2x = 9x + 3x -4 2x = 9x + 3x -4 2x +12 = 12x - 4 YES: Distribute. YES: Combine like terms! NO Go to Step 4. Step 4: Do you have variables on both sides of the equation? YES Add or subtract the terms to get all the variables on one side and all the constants on the other side. Then go to Step 6. NO Go to Step 5. Step 5: At this point, you should have a basic two-step equation to solve. Use addition or subtraction to remove any constants from the variable side of the equation. Step 6: Use multiplication or division to remove any coefficients from the variable side of the equation. 2x +12 = 12x - 4 2x 2x + 12 = 12x 2x 4 12 = 10 x = 10 x = 10x 16 = 10x YES 8/5 = x Step 7: Check your answer ⅔(x+3) +2 = 3x +⅓ (3x -4) ⅔(8/5+3) +2 = 3x +⅓ (3 8/5-4) =

81 Lesson 10: Solving Equations with Variables on Both Sides Directions: Solve each problem on your own paper. Use the Steps to Solving Equations organizer if needed. 1. 3x +2 = 2x (x +3) = 5x (x + 3) = -2(x -4) 4. -2x + 3(x -2) = -4(2x -2) 5. -5x + 4(x -2) = 3 (3-2x) 6. ⅓x -4 = 9x - ⅔ 7. ¼y 2 = 3 - ¾y 8. ½(x 5) = 3(2x -4) 9. 2y + ½(y -5) = ½(y +3) x 2(x -4) = -2(x+3) 11. 6y - ( y-4) = 2(2y +3) 12. 2x + 3(x -5) = 2(2x -6) +2x Each side of the triangle has the same length. What is the perimeter of the triangle? 4s +5 s Find the perimeter of the rectangle. 7y 6 6 y + 18 Your next assignment will be Quiz #2. This quiz will assess the following skills: Distributive Property and Equations Equations with variables on both sides Equations with Fractions Literal Equations Complete the study guide first.

82 Directions: For numbers 1-3, solve each equation. Show every step. (2 points each) 1. -2x + 4 = -4(x-6) 2. ⅓x -6 = -x - ⅔(6-x) 3. 2x + 6(x -5) = 2(2x - 15) +3x Each side of the Pentagon is equal in length. Find the perimeter of the pentagon. (2 points) 5r-3 3r + 5

83 Lesson 10: Solving Equations with Variables on Both Sides Answer Key 1. 3x +2 = 2x (x +3) = 5x 2 3x = 2x 5-2 2x +6 = 5x -2 3x = 2x -7 2x +6-6 = 5x x -2x = 2x -2x -7 2x = 5x -8 x = -7 2x -5x = 5x -5x -8-3x = x = 8/3 3. 3(x + 3) = -2(x -4) 4. -2x + 3(x -2) = -4(2x -2) 3x +9 = -2x +8-2x +3x -6 = -8x +8 3x = -2x +8-9 x 6 = -8x +8 3x = -2x -1 x +8x 6 = -8x +8x +8 3x +2x = -2x +2x -1 9x -6 = 8 5x = -1 9x 6 +6 = x = x = -1/5 x = 14/9

84 5. -5x + 4(x -2) = 3 (3-2x) 6. ⅓x -4 = 9x - ⅔ -5x +4x 8 = x 3[⅓x -4] = [ 9x - ⅔]3 -x 8 = 2x 1x -12 = 27x - 2 -x +x 8 = 2x +x 1x -1x -12 = 27x -1x -2-8 = 3x -12 = 26x = 26x /3 = x -10 = 26x /13 = x 7. ¼y 2 = 3 - ¾y 8. ½(x 5) = 3(2x -4) 4[¼y 2 ]=[ 3 - ¾y]4 2[½(x 5)] = [3(2x -4)]2 1y -8 = 12-3y 1(x -5) = 6(2x -4) 1y +3y -8 = 12-3y +3y 1x -5 = 12x -24 4y -8 = 12 1x -1x -5 = 12x -1x -24 4y = = 11x -24 4y = = 11x = 11x y = 5 19/11= x

85 9. 2y + ½(y -5) = ½(y +3) x 2(x -4) = -2(x+3) 2[2y + ½(y -5)] = [½(y +3) 4]2 3x -2x +8 = -2x -6 4y +1(y -5) = 1(y+3) -8 x +8 = -2x -6 4y +y -5 = y +3-8 x +8-8 = -2x y -5 = y -5 x = -2x -14 5y = y x +2x = -2x +2x -14 5y = y +0 3x = -14 5y y = y y 3 3 4y = x = -14/3 y = y - ( y-4) = 2(2y +3) 12. 2x + 3(x -5) = 2(2x -6) +2x -4 6y y +4 = 4y +6 2x +3x 15 = 4x 12 +2x -4 5y +4 = 4y +6 5x 15 = 4x +2x y +4-4 = 4y x -15 = 6x y = 4y +2 5x = 6x y -4y = 4y -4y +2 5x = 6x -1 y = 2 5x -6x = 6x -6x -1 -x = x = 1

86 13. Each side of the triangle has the same length. What is the perimeter of the triangle? 4s +5 s + 11 If each side of the triangle has the same length, then we can set the two sides of this triangle equal to each other in order to find the value of s. Once we find the value of s, we can substitute for one side of the triangle to find the length of the sides. 4s + 5 = s s s + 5 = s s s + 5 = 11 3s = s = 6 3s/3 = 6/3 s = 2 s = 2 s = 13 Now let s substitute to find the length of 1 side. The length of one side is 13. There are three sides, all the same length. So, 13(3) = 39 The perimeter of the triangle is Find the perimeter of the rectangle. 7y 6 6 y + 18 The lengths of a rectangle are the same dimensions. Therefore, we can set the two lengths equal to each other to solve for y. Once we find the value of y, we can substitute to find the length of the rectangle. 7y = y y y = y-y y = 18 6y/6 = 18/6 y = 3 Y = 3 Now let s substitute to find the length of the rectangle. 7y 7(3) = 21 The length of the rectangle is 21. The width is given: 6 P = 2l + 2w P = 2(21) + 2(6) P = P = 54 The perimeter of the rectangle is 54.

87 1. -2x + 4 = -4(x-6) -2x +4 = -4x +24 Distribute the -4 throughout the parenthesis -2x +4 4 = -4x Subtract 4 from both sides -2x = -4x +20 : 24-4 = 20-2x +4x = -4x+4x +20 Add 4x to both sides 2x = 20 : -2x+4x = 20 2x/2 = 20/2 Divide by 2 on both sides x = 10 : 20/2 = ⅓x -6 = -x - ⅔(6-x) 3[1/3x 6] = 3[-x -2/3(6-x)] x 18 = -3x 2(6-x) x 18 = -3x x Multiply by 3 on both sides to get rid of fraction Distribute -2 throughout the parenthesis x-18 = -3x + 2x 12 Rewrite like terms together x- 18 = -x 12 Combine like terms: -3x+2x = -x x = -x Add 18 to both sides x = -x +6 : = 6 x +x = -x+x + 6 2x = 6 Add x to both sides : x+x = 2x 2x/2 = 6/2 Divide by 2 on both sides x= 3 : 6/2 = 3

88 3. 2x + 6(x -5) = 2(2x - 15) +3x + 6 2x + 6x 30 = 4x x + 6 2x+6x-30 = 4x+3x-30x+6 8x -30 = 7x 24 8x = 7x Distribute the 6 and 2 throughout the parenthesis Rewrite with like terms together Combine like terms Add 30 to both sides 8x = 7x+6 : = 6 8x -7x = 7x-7x +6 x = 6 Subtract 7x from both sides : 8x-7x = x 4. Each side of the Pentagon is equal in length. Find the perimeter of the pentagon. (2 points) 5r-3 3r + 5 Since each side of the pentagon is equal in length, we can set the sides that we know equal to each other and solve for the variable r. Once we find the value of r, we can find the value of one side of the pentagon and then multiply by 5 to get the perimeter. 3r + 5 = 5r 3 3r 3r + 5 = 5r - 3r 3 Set the 2 sides equal to each other Subtract 3r from both sides 5 = 2r 3 : 5r- 3r = 2r = 2r Add 3 to both sides 8 = 2r : 5+3 = 8 8/2 = 2r/2 Divide by 2 on both sides 4 = r : 8/2 = 4 Since r = 4, we can substitute 4 for r. 3r + 5 = 3(4) + 5 = 17. One side is equal to 17 units, so the perimeter is 17(5) = 85 units. (Multiply by 5 because the pentagon has 5 sides. The perimeter of the pentagon is 85 units.

89 Study Guide for Concept 1:

90 Concept 2:

91 Concept 3:

92 Concept 4:

93 Equations Quiz #2 Directions: For questions 1 and 2, circle the answer that best fits the question. 1. Solve for y: -2(2y 8) = -12 a). y = -7 b). y = -1 c). y = -24 d). y = 7 2. Solve for x: 5x +2 = -2(x -7) a). x= -16/7 b). x = 4 c). x = 12/7 d). x = 16/3 3. The width of a rectangular swimming pool is 2ft less than the length. The perimeter is 30ft. What are the dimensions of the swimming pool? 4. Solve for x. Justify your answer mathematically. ¾x 7 = ¼(2x -2) +8

94 Equations Quiz #2 Answer Key Directions: For questions 1 and 2, circle the answer that best fits the question. 1. Solve for y: -2(2y 8) = -12-2(2y 8) = -12 a). y = -7-4y +16 = -12 b). y = -1-4y = c). y = -24-4y = d). y = 7 y =7 2. Solve for x: 5x +2 = -2(x -7) a). x= -16/7 5x +2 = -2(x -7) b). x = 4 5x +2- = -2x +14 c). x = 12/7 5x +2-2 = -2x d). x = 16/3 5x = -2x +12 5x +2x = -2x +2x +12 7x = x = 12/7

95 3. The width of a rectangular swimming pool is 2ft less than the length. The perimeter is 30ft. What are the dimensions of the swimming pool? P = 2L +2W P = 30 ft L= Length L-2 = Width 30 = 2L +2(L-2) L = 8.5 ft 30 = 2L +2L -4 W = L-2 so W = = 4L -4 W = = 4L The length of the swimming pool is 8.5 ft and the width is 34 = 4L 6.5 ft = L 4. Solve for x. Justify your answer mathematically. ¾x 7 = ¼(2x -2) +8 4[¾x 7] = [¼(2x -2) +8]4 Multiply BOTH sides by 4 (the denominator of the fraction.) 3x -28 = 1(2x -2) +32 3x -28 = 2x x -28 = 2x +30 3x = 2x Rewrite the problem without the parenthesis (when you distribute a 1, you get the same answer) Combine like terms. Add 28 to BOTH sides of the equation. 3x = 2x +58 3x -2x = 2x -2x +58 Subtract 2x from BOTH sides of the equation. x = 58 Justify: ¾x 7 = ¼(2x -2) +8 ¾(58) 7 = ¼(2 58-2) = 36.5

96 Lesson 11: Solving Word Problems Example 1 A computer repair store charges $45 an hour plus parts to repair computers. The parts to repair Mike s computer cost $32. The total bill was $ How many hours did they work on the computer? What I Know Define Your Variable(s). Write a Verbal Model & Substitute Solve Solution

97 Example 2 The perimeter of a flower bed is 56 feet. The length of the flower bed is 4 more than 2 times the width. Find the dimensions of the flower bed. Justify your answer. What I Know Define Your Variable(s). Write a Verbal Model & Substitute Solve Solution

98 Example 3 Kaleigh is selling pizza kits as a fundraiser. She sold 3 less pepperoni kits than supreme kits. She sold 8 less cheese kits than supreme kits. The prices of each are shown below. Cheese Kit - $12.00 Pepperoni Kit - $13.50 Supreme Kit - $15.00 Write an expression to show how many cheese kits were sold. Write an expression to show how many pepperoni kits were sold. The total sales for cheese and pepperoni kits is equal to the total sales for supreme kits. Write an equation that could be used to find how many of each type of pizza kit was sold. How many of each type of kit was sold? What I Know Define Your Variable(s). Write a Verbal Model & Substitute Solve