✏️ Practice Algebra

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📍 Overview 🔍 Observe 💡 Understand ✏️ Practice

📝 Algebra Practice Problems

📐 Quick Reference Formulas:
• Linear: ax + b = c → x = (c - b)/a
• Quadratic: x = [-b ± √(b² - 4ac)]/2a
• Slope: m = (y₂ - y₁)/(x₂ - x₁)
• Distance: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Problem B1 Beginner
Solve for x: 2x + 5 = 13

Step-by-Step Solution:

  • 2x + 5 = 13
  • Subtract 5 from both sides: 2x = 8
  • Divide both sides by 2: x = 4
✅ Answer: x = 4
Check: 2(4) + 5 = 8 + 5 = 13 ✓
Problem B2 Beginner
Solve for y: 3y - 7 = 14

Step-by-Step Solution:

  • 3y - 7 = 14
  • Add 7 to both sides: 3y = 21
  • Divide both sides by 3: y = 7
✅ Answer: y = 7
Problem B3 Beginner
Solve for x: 4(x + 2) = 20

Step-by-Step Solution:

  • 4(x + 2) = 20
  • Divide both sides by 4: x + 2 = 5
  • Subtract 2: x = 3
✅ Answer: x = 3
Alternative: Distribute first: 4x + 8 = 20 → 4x = 12 → x = 3
Problem B4 Beginner
If 5 apples cost $10, how much does one apple cost? Write and solve an equation.

Step-by-Step Solution:

  • Let a = cost of one apple
  • 5a = 10
  • Divide both sides by 5: a = 2
✅ Answer: One apple costs $2
Problem B5 Beginner
The sum of a number and 8 is 20. Find the number.

Step-by-Step Solution:

  • Let n = the number
  • n + 8 = 20
  • Subtract 8: n = 12
✅ Answer: The number is 12
Problem I1 Intermediate
Solve the system of equations:
2x + y = 10
x - y = 2

Step-by-Step Solution (Elimination):

  • Add the equations: (2x + y) + (x - y) = 10 + 2
  • 3x = 12
  • x = 4
  • Substitute into x - y = 2: 4 - y = 2
  • -y = -2
  • y = 2
✅ Answer: x = 4, y = 2
Problem I2 Intermediate
Solve the quadratic equation: x² - 5x + 6 = 0

Step-by-Step Solution (Factoring):

  • x² - 5x + 6 = 0
  • Find factors of 6 that add to -5: (-2) + (-3) = -5
  • (x - 2)(x - 3) = 0
  • x - 2 = 0 or x - 3 = 0
  • x = 2 or x = 3
✅ Answer: x = 2 or x = 3
Check: 2² - 5(2) + 6 = 4 - 10 + 6 = 0 ✓
3² - 5(3) + 6 = 9 - 15 + 6 = 0 ✓
Problem I3 Intermediate
The length of a rectangle is 3 cm more than twice its width. The perimeter is 36 cm. Find the dimensions.

Step-by-Step Solution:

  • Let w = width, then length = 2w + 3
  • Perimeter = 2(length + width) = 36
  • 2[(2w + 3) + w] = 36
  • 2(3w + 3) = 36
  • 6w + 6 = 36
  • 6w = 30
  • w = 5 cm
  • length = 2(5) + 3 = 13 cm
✅ Answer: width = 5 cm, length = 13 cm
Problem I4 Intermediate
Solve for x: 3(2x - 1) = 4x + 7

Step-by-Step Solution:

  • 3(2x - 1) = 4x + 7
  • 6x - 3 = 4x + 7
  • 6x - 4x = 7 + 3
  • 2x = 10
  • x = 5
✅ Answer: x = 5
Problem I5 Intermediate
The sum of two numbers is 28. Their difference is 4. Find the numbers.

Step-by-Step Solution:

  • Let x and y be the numbers, with x > y
  • x + y = 28
  • x - y = 4
  • Add equations: 2x = 32
  • x = 16
  • Then y = 28 - 16 = 12
✅ Answer: 16 and 12
Problem A1 Advanced
Solve using quadratic formula: 2x² - 4x - 3 = 0 (leave answer in simplest radical form)

Step-by-Step Solution:

  • a = 2, b = -4, c = -3
  • x = [-b ± √(b² - 4ac)] / 2a
  • x = [4 ± √(16 - 4×2×-3)] / 4
  • x = [4 ± √(16 + 24)] / 4
  • x = [4 ± √40] / 4
  • x = [4 ± 2√10] / 4
  • x = 1 ± (√10)/2
✅ Answer: x = 1 + √10/2 or x = 1 - √10/2
Problem A2 Advanced
Solve the inequality: -3x + 7 ≤ 16 and graph the solution on a number line.

Step-by-Step Solution:

  • -3x + 7 ≤ 16
  • Subtract 7: -3x ≤ 9
  • Divide by -3 (FLIP inequality sign): x ≥ -3
✅ Answer: x ≥ -3
Number line: ●━━━━━━━━━━→
       -3
Problem A3 Advanced
A ball is thrown upward with height given by h(t) = -5t² + 20t + 2, where h is in meters and t in seconds. Find:
(a) The height after 2 seconds
(b) The time when it hits the ground (h = 0)

Step-by-Step Solution:

  • Part (a): h(2) = -5(4) + 20(2) + 2
  • = -20 + 40 + 2 = 22 meters
  • Part (b): When h = 0: -5t² + 20t + 2 = 0
  • Multiply by -1: 5t² - 20t - 2 = 0
  • Use quadratic formula: t = [20 ± √(400 + 40)] / 10
  • t = [20 ± √440] / 10
  • t = [20 ± 2√110] / 10
  • t = 2 ± √110/5
  • t ≈ 2 ± 2.1
  • t ≈ 4.1 seconds (positive root)
✅ Answer: (a) 22 m, (b) t ≈ 4.1 seconds
Problem A4 Advanced
Solve the system:
2x + 3y = 12
4x - y = 10

Step-by-Step Solution (Substitution):

  • From second equation: y = 4x - 10
  • Substitute into first: 2x + 3(4x - 10) = 12
  • 2x + 12x - 30 = 12
  • 14x = 42
  • x = 3
  • Then y = 4(3) - 10 = 12 - 10 = 2
✅ Answer: x = 3, y = 2
Problem A5 Advanced
A boat travels 20 km upstream and 30 km downstream in a total of 5 hours. The speed of the current is 2 km/h. Find the speed of the boat in still water.

Step-by-Step Solution:

  • Let b = boat speed in still water (km/h)
  • Upstream speed = b - 2, downstream speed = b + 2
  • Time = distance/speed
  • 20/(b-2) + 30/(b+2) = 5
  • Multiply by (b-2)(b+2): 20(b+2) + 30(b-2) = 5(b² - 4)
  • 20b + 40 + 30b - 60 = 5b² - 20
  • 50b - 20 = 5b² - 20
  • 50b = 5b²
  • 5b² - 50b = 0
  • 5b(b - 10) = 0
  • b = 0 or b = 10 (b=0 not possible)
✅ Answer: Boat speed = 10 km/h

🧠 Critical Thinking Challenge

Here's a classic puzzle:

"A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?"

Most people answer 10 cents, but that's wrong! Can you set up the equations and find the correct answer?

Click for solution

Let b = ball cost, t = bat cost
t + b = 1.10
t = b + 1.00
Substitute: (b + 1.00) + b = 1.10
2b + 1.00 = 1.10
2b = 0.10
b = 0.05 ($0.05 or 5 cents)
So bat costs $1.05

📊 Problem Types Covered

Level Problems Skills Tested
🌱 Beginner B1-B5 Linear equations, basic word problems
📚 Intermediate I1-I5 Systems, quadratics, geometry applications
🚀 Advanced A1-A5 Quadratic formula, inequalities, real-world applications
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