Simplifying and Solving Equations — Solutions

Click any answer to watch the solution video.

1. Expand and simplify

   1. 3(x + 4) + 2x: 5x + 12
   2. 5(2x − 1) − 3x: 7x − 5
   3. 4(a + 3) + 2(a − 1): 6a + 10
   4. 6(b − 2) − 3(b + 1): 3b − 15
   5. 2(3m + 4) + 5(m − 2): 11m − 2
   6. 4(2p − 3) − 2(p − 5): 6p − 2
   7. 3(x + 2) + 4(2x − 1) − 5x: 6x + 2
   8. 7 − 2(3k − 4) + k: −5k + 15

2. Solve multi-step equations

   1. 3x + 7 = 22: x = 5
   2. 5x − 4 = 16: x = 4
   3. 2(x + 3) = 14: x = 4
   4. 3(2x − 1) = 15: x = 3
   5. 4x + 5 = 2x + 13: x = 4
   6. 7x − 3 = 4x + 9: x = 4
   7. 5(x − 2) = 3x + 4: x = 7
   8. 3(2x + 1) = 2(x + 9): x = 3.75 (or 15/4)

3. Equations with fractions

   1. x/3 + 5 = 9: x = 12
   2. x/4 − 2 = 3: x = 20
   3. (x + 2)/4 = 3: x = 10
   4. (2x − 1)/3 = 5: x = 8
   5. x/2 + x/3 = 10: x = 12
   6. x/4 + x/6 = 5: x = 12
   7. x/3 + x/4 = 7: x = 12
   8. x/5 − x/10 = 3: x = 30

4. Geometric equations

   1. Perimeter: 2(2x+1) + 2(x−2) = 38 → 6x − 2 = 38 → x = 40/6: 2(2x+1+x−2)=38 → 2(3x−1)=38 → 3x−1=19 → x=20/3; length = 43/3 cm, width = 14/3 cm
   2. (3x + 10) + (2x − 5) = 180: 5x + 5 = 180 → x = 35. Angles: 115° and 65°
   3. x + (2x + 15) + (x − 3) = 180: 4x + 12 = 180 → x = 42. Angles: 42°, 99°, 39°
   4. 4(2x + 3) = 8x − 4 → perimeter: Side length = perimeter ÷ 4: 2x+3 = (8x−4)/4 → 4(2x+3)=8x−4 → 8x+12=8x−4. No solution — check: perimeter of square = 4 × side. 4(2x+3) = 8x−4 means 8x+12=8x−4, impossible. The expressions are inconsistent.
   5. 4x + (5x − 12) + (3x + 24) = 360: 12x + 12 = 360 → x = 29. Angles: 116°, 133°, 111°
   6. 2(3x − 1) + (x + 7) = 31: 6x − 2 + x + 7 = 31 → 7x + 5 = 31 → x = 26/7 ≈ 3.7
   7. (3x + 15) + (2x + 5) = 180: 5x + 20 = 180 → x = 32. Angles: 111° and 69°
   8. (2x − 5) + (x + 25) = 4x + 10 (exterior angle theorem): 3x + 20 = 4x + 10 → x = 10. Exterior angle = 50°

5. Number puzzles and word problems

   1. 4n − 7 = 21: n = 7
   2. 2n + 5 = 5n − 3: n = 8/3 (or 2⅔)
   3. n + (n+1) + (n+2) = 57: 3n + 3 = 57 → n = 18. Integers: 18, 19, 20
   4. n + (n+2) + (n+4) = 78: 3n + 6 = 78 → n = 24. Integers: 24, 26, 28
   5. x + (x+8) = 46: x = 19 and x + 8 = 27
   6. 3x − x = 16: x = 8 and 3x = 24
   7. x/2 + x/3 = 20: LCM = 6: 3x + 2x = 120 → x = 24
   8. 2n − n = 48/4 = 12: n = 12

6. Real-world multi-step equations

   1. Plumber vs painter:
      1. Plumber: C = 45h + 80; Painter: C = 55h + 50
      2. 45h + 80 = 55h + 50 → 30 = 10h → h = 3 hours
      3. Plumber: 45(4)+80 = $260; Painter: 55(4)+50 = $270. Plumber is cheaper.
   2. Rectangle dimensions:
      1. Length = 2w + 5. Perimeter: 2(w + 2w + 5) = 88
      2. 2(3w + 5) = 88 → 3w + 5 = 44 → w = 13 cm, length = 31 cm
      3. Area = 13 × 31 = 403 cm²
   3. Marbles:
      1. Jordan has 2x marbles. After transfer: Sam has x + 12, Jordan has 2x − 12.
      2. x + 12 = 2x − 12 → x = 24. Sam starts with 24, Jordan with 48.
   4. Ticket sales:
      1. a + s = 80; 12a + 7s = 770
      2. Substitute s = 80 − a: 12a + 7(80−a) = 770 → 5a + 560 = 770 → a = 42. Adult tickets: 42, student tickets: 38.

7. Geometry challenge

   1. 30°, 60°, 90°  (x+2x+3x=180; x=30). Right-angled triangle.
   2. 104° and 76°  (a+b=180, a−b=28 → 2a=208 → a=104)
   3. w=10.5 m, l=37.5 m; Area=393.75 m²  (2(3w+6+w)=96 → 4w+6=48 → w=10.5)
   4. x≈60.7; six angles≈131.4°, two angles≈55.7°  (14x+50=900 → x=850/14)

8. Mobile data costs

   1. Plan A: C = 4g + 10; Plan B: C = 10g
   2. 4g + 10 = 10g → g = 5/3 ≈ 1.67 GB  (both cost ≈$16.67)
   3. Plan A: 4(8)+10=$42; Plan B: 10(8)=$80. Plan A is cheaper at 8 GB.
   4. 4g + 10 < 10g → 10 < 6g → g > 5/3 ≈ 1.67 GB. Plan A is cheaper above 1.67 GB.

9. Challenging equation problems

   1. 23, 25, 27, 29  (n+(n+2)+(n+4)+(n+6)=104 → 4n+12=104 → n=23)
   2. x=5; fraction = 6/9 = 2/3  (3(x+1)=2(2x−1) → 3x+3=4x−2 → x=5)
   3. x=13; Train A = 49 km/h, Train B = 61 km/h  (2(3x+10)+2(5x−4)=220 → 16x=208 → x=13)
   4. Ben=8, Anna=24  (A=3B; A+8=2(B+8) → 3B+8=2B+16 → B=8)

10. Extended modelling

    1. Canteen:
       1. p + s = 120; 3.50p + 2.00s = 315
       2. 50 pies and 70 sandwiches  (1.50p=75 → p=50; s=70)
       3. Sandwiches give more revenue: 70×$3.50=$245 vs 50×$2.00=$100
    2. Water tank:
       1. x=5  (12−(2x−4)=6 → 16−2x=6 → x=5)
       2. 150 minutes  (900÷6=150)
    3. w=5 cm, l=13 cm  (3w²−2w−65=0; try w=5: 75−10−65=0 ✓)
    4. 7 more weeks  (6x=240 → x=$40/wk; new rate=$55/wk; 55w≥360 → w≥6.55 → 7 weeks)