Definition:

A linear equation is an equation in which the highest power of the variable is 1. The general form is:

\( ax + b = c \) or \( ax + b = cx + d \)

Steps to Solve Linear Equations:

1. Step 1: Simplify Both Sides
   • Remove parentheses using distributive property: \( a(b + c) = ab + ac \)
   • Combine like terms on each side
   • Clear fractions by multiplying by LCD (Least Common Denominator)
2. Step 2: Move Variable Terms to One Side
   • Use addition or subtraction to move all variable terms to one side
   • Move constant terms to the opposite side
3. Step 3: Isolate the Variable
   • Divide or multiply both sides by the coefficient of the variable
   • Simplify to get \( x = \text{value} \)
4. Step 4: Check Your Solution
   • Substitute the solution back into the original equation
   • Verify that both sides are equal

Example 1: Simple Linear Equation

Solve: \( 3x + 5 = 20 \)

Step 1: Subtract 5 from both sides → \( 3x = 15 \)

Step 2: Divide both sides by 3 → \( x = 5 \)

Check: \( 3(5) + 5 = 15 + 5 = 20 \) ✓

Example 2: Variables on Both Sides

Solve: \( 5x - 7 = 2x + 11 \)

Step 1: Subtract \( 2x \) from both sides → \( 3x - 7 = 11 \)

Step 2: Add 7 to both sides → \( 3x = 18 \)

Step 3: Divide both sides by 3 → \( x = 6 \)

Check: \( 5(6) - 7 = 30 - 7 = 23 \) and \( 2(6) + 11 = 12 + 11 = 23 \) ✓

What Does "Complete the Solution" Mean?

This type of problem provides partial steps of solving an equation. You need to identify missing steps, fill in blank spaces, or complete the final steps to reach the solution.

Skills Required:

• Identify Operations: Recognize what operation was performed at each step
• Apply Properties: Use the correct property of equality
• Follow Logic: Understand the sequence of steps in solving equations
• Verify Steps: Check that each step is mathematically correct

Example: Complete the Solution

Solve: \( 4(x - 3) = 2x + 6 \)

Answer: \( x = 9 \)

Common Mistakes to Avoid:

• Forgetting to distribute negative signs: \( -(x - 3) = -x + 3 \)
• Not performing the same operation on both sides
• Incorrectly combining unlike terms
• Sign errors when moving terms across the equal sign

Translating Words to Equations:

Word problems require converting verbal descriptions into mathematical equations.

Step-by-Step Strategy:

1. Read Carefully: Understand what the problem is asking and identify given information
2. Define Variables: Let \( x \) (or another letter) represent the unknown quantity
3. Translate to Equation: Convert the word problem into a mathematical equation using key phrases
4. Solve the Equation: Use algebraic methods to find the value of the variable
5. Answer the Question: Write your answer in the context of the problem
6. Check Your Answer: Verify that your solution makes sense in the context

Key Phrases and Their Math Meanings:

Example 1: Number Problem

"Five more than three times a number is 26. Find the number."

Let \( x \) = the unknown number

Translate: "Three times a number" = \( 3x \)

"Five more than..." = \( 3x + 5 \)

Equation: \( 3x + 5 = 26 \)

Solve: \( 3x = 21 \) → \( x = 7 \)

Answer: The number is 7

Example 2: Age Problem

"Maria is 8 years older than her brother. The sum of their ages is 32. How old is Maria?"

Let \( x \) = brother's age

Then \( x + 8 \) = Maria's age

Equation: \( x + (x + 8) = 32 \)

Simplify: \( 2x + 8 = 32 \)

Solve: \( 2x = 24 \) → \( x = 12 \)

Answer: Maria is \( 12 + 8 = 20 \) years old

Example 3: Cost Problem

"A plumber charges $75 for a visit plus $40 per hour. If the total bill was $235, how many hours did the plumber work?"

Let \( h \) = number of hours worked

Equation: \( 75 + 40h = 235 \)

Solve: \( 40h = 160 \) → \( h = 4 \)

Answer: The plumber worked 4 hours

Definition:

Rearranging (or solving for a variable) means isolating one variable in terms of the other variables. These are also called literal equations or formulas.

Steps to Rearrange Multi-Variable Equations:

1. Identify the Variable to Isolate: Determine which variable you need to solve for (make it the subject)
2. Treat Other Variables as Constants: All other variables are treated like numbers
3. Use Inverse Operations: Apply the same algebraic steps as solving regular equations:
   • Add or subtract terms from both sides
   • Multiply or divide both sides by expressions
   • Use distributive property when needed
4. Factor if Necessary: If the variable appears multiple times, factor it out
5. Simplify the Result: Write the variable alone on one side of the equation

Example 1: Simple Rearrangement

Solve for x: \( y = mx + b \)

Step 1: Subtract \( b \) from both sides → \( y - b = mx \)

Step 2: Divide both sides by \( m \) → \( \frac{y - b}{m} = x \)

Answer: \( x = \frac{y - b}{m} \)

Example 2: Area Formula

Solve for w: \( A = lw \) (Area of rectangle)

Step 1: Divide both sides by \( l \) → \( \frac{A}{l} = w \)

Answer: \( w = \frac{A}{l} \)

Example 3: Perimeter Formula

Solve for w: \( P = 2l + 2w \) (Perimeter of rectangle)

Step 1: Subtract \( 2l \) from both sides → \( P - 2l = 2w \)

Step 2: Divide both sides by 2 → \( \frac{P - 2l}{2} = w \)

Answer: \( w = \frac{P - 2l}{2} \)

Example 4: Temperature Conversion

Solve for C: \( F = \frac{9}{5}C + 32 \)

Step 1: Subtract 32 from both sides → \( F - 32 = \frac{9}{5}C \)

Step 2: Multiply both sides by \( \frac{5}{9} \) → \( \frac{5}{9}(F - 32) = C \)

Answer: \( C = \frac{5}{9}(F - 32) \)

Example 5: Factoring Required

Solve for x: \( ax + bx = c \)

Step 1: Factor out \( x \) → \( x(a + b) = c \)

Step 2: Divide both sides by \( (a + b) \) → \( x = \frac{c}{a + b} \)

Answer: \( x = \frac{c}{a + b} \)

• Solving Linear Equations: Simplify → Move variables to one side → Isolate variable → Check solution
• Completing Solutions: Identify operations, follow logical steps, verify each step is correct
• Word Problems: Read carefully → Define variable → Translate to equation → Solve → Answer in context
• Rearranging Formulas: Treat other variables as constants → Use inverse operations → Factor if needed

1. One Solution (Most Common)

Example: \( 2x + 3 = 7 \) → \( x = 2 \)

2. No Solution (Contradiction)

Example: \( x + 3 = x + 5 \) → \( 3 = 5 \) (False statement, no solution)

3. Infinite Solutions (Identity)

Example: \( 2x + 4 = 2(x + 2) \) → \( 2x + 4 = 2x + 4 \) (Always true, all real numbers are solutions)