Solving Linear Equations

What is a Linear Equation?

A linear equation is an algebraic equation where the highest power of the variable is 1. The graph of a linear equation is always a straight line. Linear equations have the general form:

ax + b = c

Where:

• x is the variable (unknown)
• a, b, and c are constants (known numbers)
• a ≠ 0

Examples of linear equations:

• 2x + 5 = 13
• 3x - 7 = 20
• x/4 + 3 = 7
• 5(x - 2) = 15

Goal of Solving Equations

The goal is to isolate the variable on one side of the equation. This means getting x (or whatever variable) alone, with a coefficient of 1.

We want to transform the equation into: x = (some number)

Properties of Equality

These properties allow us to manipulate equations while keeping them balanced:

1. Addition Property of Equality If a = b, then a + c = b + c You can add the same number to both sides.

2. Subtraction Property of Equality If a = b, then a - c = b - c You can subtract the same number from both sides.

3. Multiplication Property of Equality If a = b, then a × c = b × c (where c ≠ 0) You can multiply both sides by the same non-zero number.

4. Division Property of Equality If a = b, then a ÷ c = b ÷ c (where c ≠ 0) You can divide both sides by the same non-zero number.

Solving One-Step Equations

Addition/Subtraction Equations:

Example 1: x + 7 = 12 Subtract 7 from both sides: x + 7 - 7 = 12 - 7 x = 5

Example 2: x - 4 = 9 Add 4 to both sides: x - 4 + 4 = 9 + 4 x = 13

Multiplication/Division Equations:

Example 3: 3x = 15 Divide both sides by 3: 3x/3 = 15/3 x = 5

Example 4: x/2 = 8 Multiply both sides by 2: 2 × (x/2) = 2 × 8 x = 16

Solving Two-Step Equations

Two-step equations require two operations to solve. Use reverse order of operations - undo addition/subtraction first, then multiplication/division.

General Strategy:

1. Undo addition or subtraction
2. Undo multiplication or division

Example 1: 2x + 5 = 13 Step 1: Subtract 5 from both sides 2x + 5 - 5 = 13 - 5 2x = 8

Step 2: Divide both sides by 2 2x/2 = 8/2 x = 4

Check: 2(4) + 5 = 8 + 5 = 13 ✓

Example 2: 3x - 7 = 20 Step 1: Add 7 to both sides 3x - 7 + 7 = 20 + 7 3x = 27

Step 2: Divide both sides by 3 x = 9

Example 3: x/4 + 3 = 7 Step 1: Subtract 3 from both sides x/4 = 4

Step 2: Multiply both sides by 4 x = 16

Solving Multi-Step Equations

For more complex equations, follow this order:

Step 1: Simplify each side (distribute, combine like terms) Step 2: Get all variable terms on one side Step 3: Get all constants on the other side Step 4: Isolate the variable

Example 1: 5(x - 2) = 15 Step 1: Distribute 5 5x - 10 = 15

Step 2: Add 10 to both sides 5x = 25

Step 3: Divide by 5 x = 5

Example 2: 3x + 7 = x + 19 Step 1: Subtract x from both sides 2x + 7 = 19

Step 2: Subtract 7 from both sides 2x = 12

Step 3: Divide by 2 x = 6

Example 3: 2(x + 3) - 5 = 4x - 7 Step 1: Distribute 2x + 6 - 5 = 4x - 7 2x + 1 = 4x - 7

Step 2: Subtract 2x from both sides 1 = 2x - 7

Step 3: Add 7 to both sides 8 = 2x

Step 4: Divide by 2 4 = x, or x = 4

Equations with Variables on Both Sides

When variables appear on both sides, collect all variable terms on one side and all constants on the other.

Example: 5x - 3 = 2x + 12 Step 1: Subtract 2x from both sides 3x - 3 = 12

Step 2: Add 3 to both sides 3x = 15

Step 3: Divide by 3 x = 5

Equations with Fractions

Method 1: Clear fractions by multiplying by LCD

Example: (x/3) + (x/4) = 7 LCD = 12 Multiply everything by 12: 12(x/3) + 12(x/4) = 12(7) 4x + 3x = 84 7x = 84 x = 12

Method 2: Work with fractions directly

Example: (2x + 1)/3 = 5 Multiply both sides by 3: 2x + 1 = 15 2x = 14 x = 7

Checking Your Solution

Always check by substituting your answer back into the original equation.

Example: If solving 3x + 5 = 14 gives x = 3 Check: 3(3) + 5 = 9 + 5 = 14 ✓

Special Cases

Identity (Infinitely Many Solutions): When you get a statement that's always true (like 0 = 0 or 5 = 5) Example: 2(x + 3) = 2x + 6 After simplifying: 6 = 6 (true for all x)

Contradiction (No Solution): When you get a false statement (like 0 = 5) Example: 2(x + 3) = 2x + 8 After simplifying: 6 = 8 (never true)

Common Mistakes to Avoid

1. Not doing the same thing to both sides Wrong: If 2x = 10, x = 10 - 2 = 8 Right: Divide both sides by 2, x = 5

2. Forgetting to distribute Wrong: 3(x + 2) = 3x + 2 Right: 3(x + 2) = 3x + 6

3. Sign errors when moving terms 2x + 5 = 13 Remember: 2x = 13 - 5, not 13 + 5

4. Dividing by zero Never divide both sides by 0

5. Not checking the solution Always substitute back to verify!

Real-World Applications

Linear equations appear in many real-life situations:

Example 1: Age Problems "Sarah is 5 years older than Tom. Together their ages sum to 27. How old is Tom?" Let x = Tom's age x + (x + 5) = 27 2x + 5 = 27 x = 11 (Tom is 11)

Example 2: Money Problems "You have $50. After buying some items at$7 each, you have $15 left. How many items did you buy?" 50 - 7x = 15 7x = 35 x = 5 items

Example 3: Geometry "The perimeter of a rectangle is 40 cm. The length is 3 cm more than the width. Find the dimensions." Let w = width Perimeter: 2w + 2(w + 3) = 40 4w + 6 = 40 w = 8.5 cm

Problem-Solving Strategy

1. Read the problem carefully
2. Identify what you're solving for (define variable)
3. Write an equation
4. Solve the equation
5. Check if the answer makes sense
6. Answer the question in words

Quick Reference

| Type | Example | First Step | |------|---------|------------| | x + 5 = 12 | One-step | Subtract 5 | | 3x = 15 | One-step | Divide by 3 | | 2x + 5 = 13 | Two-step | Subtract 5 | | 3(x - 2) = 15 | Multi-step | Distribute 3 | | 5x - 3 = 2x + 12 | Variables both sides | Collect variables |

Practice Tips

• Always show your work step-by-step
• Check your solution by substituting back
• Keep equations balanced (what you do to one side, do to the other)
• Work in reverse order of operations
• Simplify before solving when possible
• Be careful with negative signs