Foundations: Linear Equations and Algebraic Manipulation

TL;DR

You'll master isolating variables and manipulating equations—the core skills for solving systems by substitution. These techniques let you rearrange any linear equation to solve for any variable. Think of it as the foundation that makes substitution possible.

1. The Mental Model

Linear equations are like balanced scales—whatever you do to one side, you must do to the other. Your job is to "unwrap" the variable you want by undoing operations in reverse order. That's the whole idea.

2. The Core Material

2.1 The Isolation Process

When you solve for a variable, you're essentially undoing everything that's been done to it. Look at 3x + 7 = 22. The variable x has been multiplied by 3, then had 7 added to it. To isolate x, you reverse these operations in opposite order:

First, undo the addition: 3x + 7 - 7 = 22 - 7, so 3x = 15
Then, undo the multiplication: 3x ÷ 3 = 15 ÷ 3, so x = 5

The key insight? You always work backwards through the order of operations. If the variable was multiplied then something was added, you subtract first, then divide.

This reverse-order thinking becomes crucial when you're solving for one variable in terms of another. Take 2x + 3y = 12. To solve for x in terms of y:
- Subtract 3y from both sides: 2x = 12 - 3y
- Divide by 2: x = 6 - (3/2)y

Now x is expressed entirely in terms of y, which is exactly what you need for substitution.

2.2 Fraction and Decimal Handling

Don't let fractions intimidate you—they follow the same rules. If you have x/4 + 5 = 8:
- Subtract 5: x/4 = 3
- Multiply by 4: x = 12

When solving for a variable creates fractions, embrace them. From 3x + 2y = 7, solving for y gives:
y = (7 - 3x)/2

You could also write this as y = 7/2 - (3/2)x. Both forms are correct, but the first is often cleaner for substitution.

Decimals work identically. For 0.5x + 1.2 = 4.7:
- Subtract 1.2: 0.5x = 3.5
- Divide by 0.5: x = 7

2.3 Handling Variables on Both Sides

Sometimes you'll encounter equations like 4x + 3 = 2x + 11. Here's your strategy:

1. Get all x terms on one side by subtracting 2x: 2x + 3 = 11
2. Get constants on the other side by subtracting 3: 2x = 8
3. Divide: x = 4

This skill becomes essential when you substitute one equation into another and end up with the variable appearing multiple times.

When solving for one variable in terms of another with variables on both sides, like 2x + y = x + 6:
- Subtract x from both sides: x + y = 6
- Subtract y: x = 6 - y

The process is identical—collect like terms, then isolate.

flowchart TD
    A["Original equation: 3x + 7 = 22"] --> B["Identify operations on variable"]
    B --> C["Multiplication by 3, then addition of 7"]
    C --> D["Reverse order: subtract 7 first"]
    D --> E["3x = 15"]
    E --> F["Then divide by 3"]
    F --> G["x = 5"]

3. Worked Example

Let's solve the system:
- Equation 1: 2x + 3y = 16
- Equation 2: x - y = 2

Step 1: Choose which equation to solve for which variable. Equation 2 looks simpler, and x has coefficient 1, so let's solve for x.

From x - y = 2:
Add y to both sides: x = 2 + y

Step 2: Substitute this expression for x into Equation 1.
2x + 3y = 16 becomes:
2(2 + y) + 3y = 16

Step 3: Solve for y.
Distribute: 4 + 2y + 3y = 16
Combine like terms: 4 + 5y = 16
Subtract 4: 5y = 12
Divide by 5: y = 12/5 or y = 2.4

Step 4: Substitute back to find x.
x = 2 + y = 2 + 12/5 = 10/5 + 12/5 = 22/5 or x = 4.4

Check: 2(4.4) + 3(2.4) = 8.8 + 7.2 = 16 ✓
4.4 - 2.4 = 2 ✓

4. Key Takeaways

4.1 Most Important Concepts

• Balance principle: Whatever you do to one side of an equation, do to the other
• Reverse order: Undo operations in the opposite order they were applied
• Variable isolation: Get the target variable alone on one side by systematically eliminating everything else
• Substitution readiness: Solve for variables in terms of other variables, not just numbers
• Fraction acceptance: Don't avoid fractions—they're often the correct answer
• Like terms: Combine terms with the same variable before isolating
• Strategic choice: Pick the easiest equation and variable combination to minimize work

4.2 Common Misconceptions

• "I must eliminate fractions immediately" → Fractions are fine and often simpler than forcing whole numbers
• "I can only solve for numbers" → You can solve for variables in terms of other variables
• "Operations can be done in any order" → You must reverse the order of operations systematically
• "Both sides don't need the same operation" → The balance principle is non-negotiable

4.3 Compare & Contrast

5. Now Try It

Solve this system using substitution: 3x + 2y = 13 and x + 4y = 11. First, practice the algebraic manipulation by solving the second equation for x, then substitute into the first equation and solve completely for both variables.

Success looks like: You get specific numerical values for both x and y, and when you substitute them back into both original equations, both equations are satisfied.