Unit I: Solving Linear Equations

The Problem: Beyond Simple Addition

In the last chapter, we learned the golden rule of algebra: whatever you do to one side, you must do to the other. We used this to solve equations like x + 3 = 7 by removing blocks from a scale.

But what about an equation like 2x + 3 = 11? We can't just remove 'x' from the left side anymore. We need a more powerful set of tools. The "Aha!" moment is realizing that every mathematical operation has an inverse—an opposite operation that undoes it.

• The inverse of Addition is Subtraction.
• The inverse of Multiplication is Division.

To solve any linear equation, we just apply these inverse operations to both sides until 'x' is left all by itself.

The Order of Operations (PEMDAS)

To understand how to solve an equation, you first need to know how they are built. Expressions are evaluated in a specific order, commonly remembered by the acronym PEMDAS:

1. Parentheses
2. Exponents
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

For the expression 2x + 3, PEMDAS tells us you first multiply 'x' by 2, and *then* you add 3.

The "Aha!" Moment: Reversing the Story of 'x'

Think of the expression 2x + 3 as a short story describing what happened to 'x':

1. First, 'x' was multiplied by 2.
2. Then, 3 was added to that result.

To solve the equation and find the original value of 'x', we must reverse this story by undoing each step in the opposite order. This is why we use "PEMDAS in reverse" (or SADMEP).

1. First, we undo the last step (the addition) by subtracting 3 from both sides.
2. Then, we undo the first step (the multiplication) by dividing by 2 on both sides.

This "reverse story" method is the key to isolating 'x'. Use the interactive solver below to walk through this process.

The Final Challenge: Variables on Both Sides

What happens when the mystery box 'x' appears on both sides of the scale? Consider the equation 5x - 4 = 2x + 8.

This looks complicated, but the goal is the same: simplify the problem until it's one we already know how to solve. The strategy has two main parts:

1. Consolidate the 'x' terms: Use the Golden Rule to get all the 'x' terms together on one side of the equation. A good habit is to eliminate the *smaller* 'x' term to avoid negative numbers. In this case, we would subtract 2x from both sides.
2. Isolate the 'x' term: Once all the 'x's are on one side, the equation becomes a simple two-step problem that you can solve using the "reverse story" method.

By doing this, you turn a complex problem into a familiar one. Let's walk through 5x - 4 = 2x + 8 using the solver below.

The 4-Step Method to Solve Any Linear Equation

1. Distribute: If there are parentheses, apply the distributive property first.
2. Combine Like Terms: Simplify each side of the equation by combining 'x' terms and constant terms.
3. Move Variables: Add or subtract terms to get all the variables on one side of the equation.
4. Isolate the Variable: Use inverse operations (addition/subtraction, then multiplication/division) to solve for 'x'.