⚖Basic Linear Equations
Linear equations are like balancing scales—what you do to one side, you must do to the other! Learn how to solve equations and find the value of unknown variables with simple, logical steps.
🎯 What is a Linear Equation?
A linear equation is an equation where the variable appears to the first power (no exponents like $x^2$). It forms a straight line when graphed, hence the name "linear"!
Examples of Linear Equations
- $x + 5 = 12$ ✅
- $3x - 7 = 14$ ✅
- $\frac{x}{2} + 3 = 9$ ✅
- $x^2 + 5 = 9$ ❌ (not linear because of $x^2$)
⚖️ The Balance Method
Think of an equation as a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced!
Solving: $x + 5 = 12$
Step 1: We want to isolate $x$ on one side
Step 2: Subtract 5 from both sides:
$$x + 5 - 5 = 12 - 5$$
Step 3: Simplify:
$$x = 7$$
✨ Check your answer: $7 + 5 = 12$ ✅
➕ Addition and Subtraction
When you have $x + a = b$, subtract $a$ from both sides.
When you have $x - a = b$, add $a$ to both sides.
Example 1: $x + 8 = 15$
Subtract 8 from both sides: $x = 15 - 8 = 7$
Example 2: $x - 3 = 10$
Add 3 to both sides: $x = 10 + 3 = 13$
✖️➗ Multiplication and Division
When you have $ax = b$, divide both sides by $a$.
When you have $\frac{x}{a} = b$, multiply both sides by $a$.
Example 1: $3x = 21$
Divide both sides by 3:
$$x = \frac{21}{3} = 7$$
Example 2: $\frac{x}{4} = 5$
Multiply both sides by 4:
$$x = 5 \times 4 = 20$$
🔄 Multi-Step Equations
Sometimes you need to use multiple operations to solve an equation. Work step by step!
Solving: $2x + 3 = 11$
Step 1: Subtract 3 from both sides
$2x = 11 - 3 = 8$
Step 2: Divide both sides by 2
$$x = 4$$ ✨
Check: $2(4) + 3 = 8 + 3 = 11$ ✅
🌟 Real-World Applications
- 💰 Shopping: "I bought 3 shirts for $45 total. How much was each shirt?" → $3x = 45$
- 🚗 Travel: "A car travels 60 mph for $x$ hours and goes 240 miles" → $60x = 240$
- 🎂 Age Problems: "Sarah is 5 years older than Tom. Sarah is 12" → $x + 5 = 12$
🎯 Practice Questions
Solve these equations to master your skills!
🔥 Challenge Questions
Ready for a bigger challenge?