Solving Linear Equations

TL;DR

Solving linear equations is like balancing a scale: whatever you do to one side, you must do to the other to keep it balanced. Your goal is to isolate the variable (like 'x') on one side of the equation. We use inverse operations to move numbers and variables until the variable stands alone.

1. The Mental Model

Think of a linear equation as a perfectly balanced seesaw. Each side of the equation has the same "weight" or value. To keep the seesaw balanced, any operation you perform on one side (like adding or subtracting weight) must also be done on the exact other side.

2. The Core Material

Solving a linear equation means finding the value of the unknown variable that makes the equation true. Most often, this variable is x.

The Goal: Isolate the Variable

Your primary goal is to get the variable by itself on one side of the equals sign. For example, you want to transform x + 5 = 10 into x = 5.

Inverse Operations are Your Tools

To move numbers or terms away from the variable, you use inverse operations. These are operations that "undo" each other:

• Addition undoes subtraction (and vice-versa).
• Multiplication undoes division (and vice-versa).

Steps to Solve Linear Equations

Here's a general process flow for solving linear equations:

graph TD
    A["Start with the equation"] --> B{"Are there parentheses?"}
    B -- Yes --> C["Distribute to remove parentheses"]
    C --> D{"Are there like terms on the same side?"}
    B -- No --> D
    D -- Yes --> E["Combine like terms on each side"]
    E --> F{"Is the variable on both sides?"}
    D -- No --> F
    F -- Yes --> G["Move all variable terms to one side (using addition/subtraction)"]
    G --> H["Move all constant terms to the other side (using addition/subtraction)"]
    F -- No --> H
    H --> I{"Is the variable multiplied/divided by a number?"}
    I -- Yes --> J["Multiply or divide to isolate the variable"]
    J --> K["Solution: x = (number)"]
    I -- No --> K

Let's break down those steps with examples:

Step 1: Simplify Each Side (Parentheses and Like Terms)

Before you start moving terms across the equals sign, make sure each side of the equation is as simple as possible.

• Distribute: If you have parentheses, multiply the number outside by everything inside.
  • Example: 3(x + 2) = 15 becomes 3x + 6 = 15
• Combine Like Terms: If you have multiple x terms or multiple constant numbers on the same side of the equals sign, combine them.
  • Example: 2x + 5 + 3x = 20 becomes 5x + 5 = 20

Step 2: Get Variables to One Side

If the variable (like x) appears on both sides of the equation, you need to gather all variable terms on one side. Choose the side that will keep the variable coefficient positive, if possible.

• Example: 5x - 3 = 2x + 9
  • Subtract 2x from both sides:
    5x - 2x - 3 = 2x - 2x + 9
3x - 3 = 9

Step 3: Get Constants to the Other Side

Now that all variable terms are on one side, move all the constant numbers (numbers without a variable) to the other side.

• Continuing from above: 3x - 3 = 9
  • Add 3 to both sides:
    3x - 3 + 3 = 9 + 3
3x = 12

Step 4: Isolate the Variable (Multiply or Divide)

The last step is usually to get x completely alone. If x is being multiplied or divided by a number, use the inverse operation.

• Continuing from above: 3x = 12
  • Divide both sides by 3:
    3x / 3 = 12 / 3
x = 4

Step 5: Check Your Solution

Always, always, always check your answer! Substitute your solution back into the original equation to make sure both sides are equal.

• Check x = 4 in 5x - 3 = 2x + 9:
  5(4) - 3 = 2(4) + 9
20 - 3 = 8 + 9
17 = 17
  Since 17 = 17 is true, your solution x = 4 is correct!

3. Worked Example

Let's solve the equation: 4(2x - 1) + 5 = 3x + 15

1. Distribute to remove parentheses:
   8x - 4 + 5 = 3x + 15

2. Combine like terms on the left side:
   8x + 1 = 3x + 15

3. Move all variable terms to one side. Let's move 3x to the left side by subtracting 3x from both sides:
   8x - 3x + 1 = 3x - 3x + 15
5x + 1 = 15

4. Move all constant terms to the other side. Let's move 1 to the right side by subtracting 1 from both sides:
   5x + 1 - 1 = 15 - 1
5x = 14

5. Multiply or divide to isolate the variable. Divide both sides by 5:
   5x / 5 = 14 / 5
x = 14/5 (or x = 2.8)

6. Check your solution in the original equation 4(2x - 1) + 5 = 3x + 15 using x = 14/5:
   Left Side: 4(2 * (14/5) - 1) + 5
= 4(28/5 - 5/5) + 5
= 4(23/5) + 5
= 92/5 + 25/5
= 117/5

   Right Side: 3 * (14/5) + 15
= 42/5 + 75/5
= 117/5

   Since 117/5 = 117/5, the solution x = 14/5 is correct.

4. Key Takeaways

• Always aim to get the variable (like x) by itself on one side of the equation.
• Use inverse operations to "undo" additions/subtractions and multiplications/divisions.
• Whatever operation you perform on one side of the equation, you must perform on the other side.
• Simplify each side of the equation (distribute, combine like terms) before moving terms across the equals sign.
• Move variable terms to one side, then constant terms to the other, then divide/multiply.
• Crucially, always check your solution by plugging it back into the original equation.

Common Mistakes to Avoid:
- Forgetting to apply an operation to both sides of the equation.
- Making sign errors when moving terms (e.g., subtracting 3x from one side but adding it to the other).
- Incorrectly distributing a negative number or a number outside parentheses.
- Combining terms that aren't "like terms" (e.g., 2x + 3 does not equal 5x).

5. Now Try It

Solve the following equation and check your answer:

2(3y + 4) - 10 = 5y + 7

What to do:
1. Follow the steps outlined in "The Core Material".
2. Show each step clearly.
3. Verify your solution by substituting it back into the original equation.

What success looks like:
You should arrive at a single numerical value for y, and when you substitute that value back into the original equation, both sides should be equal.