CodeMathFusion
☰

Basic Linear Equations

1. What is a Linear Equation?

A linear equation is a simple equation that forms a straight line when graphed. It usually contains a variable (like x or y) and does not have exponents or square roots.

Real-Life Example:

Imagine you are renting a bicycle at a shop. The shop charges a $5 fixed fee plus $2 per hour. If you rent the bicycle for h hours, your total cost is:

Total Cost = 5 + 2h

This is a linear equation because it follows a straight-line relationship.

2. Understanding One-Step Equations

A one-step equation is an equation that can be solved using just one mathematical operation (addition, subtraction, multiplication, or division).

Example: x + 5 = 12
To find x, subtract 5 from both sides:
x = 12 - 5
x = 7

This means x = 7 is the solution!

3. Solving One-Step Equations (Addition & Subtraction)

Addition Example:
x - 3 = 10
Add 3 to both sides:
x = 10 + 3
x = 13

Subtraction Example:
y + 8 = 15
Subtract 8 from both sides:
y = 15 - 8
y = 7

4. Solving One-Step Equations (Multiplication & Division)

Multiplication Example:
3x = 12
Divide both sides by 3:
x = 12 / 3
x = 4

Division Example:
y / 5 = 6
Multiply both sides by 5:
y = 6 * 5
y = 30

5. Understanding Two-Step Equations

A two-step equation requires two operations to find the solution.

Example: 2x + 3 = 11
Step 1: Subtract 3 from both sides: 2x = 11 - 3 = 8
Step 2: Divide both sides by 2: x = 8 / 2 = 4
So, x = 4 is the solution!

6. Checking Your Solutions

Always plug your answer back into the original equation to ensure it is correct.

Example: Solve 5x - 2 = 13
Step 1: Add 2 to both sides: 5x = 15
Step 2: Divide by 5: x = 15 / 5 = 3
Check: 5(3) - 2 = 15 - 2 = 13 ✔

7. Writing Equations from Word Problems

Algebra is useful in real life! Let's learn how to turn word problems into equations.

Example 1:
Emma has $20 and spends $3 per day. How many days (d) until she has $5 left?
Equation: 20 - 3d = 5

Example 2:
A taxi charges $4 plus $2 per mile. If the total cost is $12, how many miles (m) did you travel?
Equation: 4 + 2m = 12

8. Real-Life Applications of Linear Equations

  • ✔ Budgeting: Tracking expenses using equations.
  • ✔ Sports: Calculating speed and distance.
  • ✔ Business: Predicting profits based on sales.
  • ✔ Cooking: Adjusting recipes for more people.

Practice Questions (Basic Level) 🎯

  1. Solve for x: x + 7 = 15
  2. Solve for y: y - 4 = 12
  3. Solve for z: 3z = 21
  4. Solve for x: x ÷ 5 = 6
  5. Solve for m: 2m + 5 = 13
  6. Solve for b: 4b - 3 = 9
  7. Solve for p: (p / 4) + 2 = 10
  8. Write an equation: James earns $5 per hour and made $30. How many hours (h) did he work?
  9. A gym charges $10 monthly fee plus $3 per session. Write an equation for total cost (c) after s sessions.
  10. The sum of a number and 9 is 15. Write an equation and solve for the number.

Want a Bigger Challenge? 🤔🔥

  1. Solve for x: 5(x - 2) = 20
  2. Solve for y: (y / 3) + 4 = 10
  3. Jack spends $20 on movie tickets and buys p popcorns at $4 each. If he spends $40 total, write and solve an equation for p.
  4. The sum of three consecutive numbers is 36. Find the numbers.
  5. A cell phone plan costs $15 per month plus $0.10 per text. If your total bill was $35, how many texts (t) did you send?

11. Summary

  • ✅ Linear equations help us solve real-life problems.
  • ✅ One-step equations involve one operation (add, subtract, multiply, or divide).
  • ✅ Two-step equations require two operations to solve.
  • ✅ Checking solutions helps confirm answers.
  • ✅ Word problems can be written as equations.
  • ✅ Algebra is used in everyday life—from shopping to budgeting to sports!

🎉 Great job! Now that you understand Basic Linear Equations, you're ready for the next topic: Introduction to Inequalities! 🚀

Next Topic →