Foundations of Linear Equations in One Variable

TL;DR

Linear equations in one variable are mathematical statements where you're finding the value that makes both sides equal. They follow the pattern ax + b = c, where you isolate x through inverse operations. Master these foundations and you'll solve any linear equation systematically.

1. The Mental Model

Think of a linear equation as a balance scale that's perfectly even. Whatever you do to one side, you must do to the other to keep it balanced. Your goal is to get the variable alone on one side by undoing operations in reverse order. That's the whole idea.

2. The Core Material

What Makes an Equation Linear

A linear equation in one variable has exactly one unknown (usually x) raised only to the first power. No x², no √x, no fractions with x in the denominator. The general form is ax + b = c, where a, b, and c are numbers and a ≠ 0.

Examples of linear equations:
- 3x + 7 = 22
- -2x = 14
- x/4 - 3 = 5
- 0.5x + 1.2 = 3.7

Non-linear equations you'll encounter later:
- x² + 3 = 12 (quadratic)
- 2/x = 8 (rational)
- √x = 4 (radical)

The Golden Rule: Balance and Inverse Operations

Every equation is a statement that two expressions are equal. To solve it, you'll use inverse operations to isolate the variable. Here's your toolkit:

Addition ↔ Subtraction: If something's added to x, subtract it from both sides. If something's subtracted from x, add it to both sides.

Multiplication ↔ Division: If x is multiplied by a number, divide both sides by that number. If x is divided by a number, multiply both sides by that number.

Order matters: Always undo operations in reverse order of operations (PEMDAS backwards). Handle addition and subtraction first, then multiplication and division.

The Four-Step Solution Process

1. Simplify each side if needed (combine like terms, distribute, clear fractions)
2. Move variable terms to one side using addition/subtraction
3. Move constant terms to the other side using addition/subtraction
4. Isolate the variable using multiplication/division

Let's see this with 2(x + 3) - 5 = 13:

Step 1: Distribute and simplify
2x + 6 - 5 = 13
2x + 1 = 13

Step 2: Variable terms are already on the left

Step 3: Subtract 1 from both sides
2x = 12

Step 4: Divide both sides by 2
x = 6

Checking Your Answer

Always substitute your answer back into the original equation. If both sides equal the same number, you're correct. Using x = 6:

2(6

Linear Equations in Two Variables: Graphing and Properties

TL;DR

Linear equations in two variables create straight lines when graphed. You'll learn to identify slope, y-intercept, and different forms of these equations. Every straight line represents infinitely many solutions to the equation.

1. The Mental Model

A linear equation with two variables is like a recipe that connects x and y values. For every x you pick, the equation tells you exactly what y must be. Plot enough of these (x,y) pairs and you'll always get a perfectly straight line. That's the whole idea.

2. The Core Material

Standard Form and Slope-Intercept Form

Linear equations come in different forms, but they're all describing the same straight line. The most common form you'll work with is slope-intercept form: y = mx + b.

In this form, m is the slope (how steep the line is) and b is the y-intercept (where the line crosses the y-axis). For example, y = 3x + 2 has a slope of 3 and crosses the y-axis at (0, 2).

Standard form looks like Ax + By = C, where A, B, and C are integers. The equation 2x + 3y = 12 is in standard form. You can convert between forms using algebra. To go from standard to slope-intercept, solve for y:
2x + 3y = 12
3y = -2x + 12
y = -⅔x + 4

Now you can see the slope is -⅔ and the y-intercept is 4.

Understanding Slope

Slope measures how much y changes when x increases by 1. It's the "rise over run." A slope of 3 means for every step right, you go up 3 steps. A slope of -½ means for every 2 steps right, you go down 1 step.

You can calculate slope between any two points (x₁, y₁) and (x₂, y₂) using: m = (y₂ - y₁)/(x₂ - x₁)

Positive slopes go upward from left to right. Negative slopes go downward. A slope of zero is horizontal. An undefined slope (division by zero) is vertical.

Graphing Strategies

To graph a linear equation, you need at least two points. Here are the fastest methods:

Method 1: Use slope and y-intercept. Start at the y-intercept, then use the slope to find your next point. If y = 2x - 1, start at (0, -1), then go right 1 and up 2 to reach (1, 1).

Method 2: Find intercepts. Set x = 0 to find the y-intercept, then set y = 0 to find the x-intercept. For 2x + 3y = 12: when x = 0, y = 4. When y = 0, x = 6. Plot (0, 4) and (6, 0), then connect them.

Method 3: Make a table. Pick any x-values, calculate the corresponding y-values, and plot those points.

```mermaid
graph LR
A["Pick two points"] --> B["Plot on

Systems of Linear Equations in Two Variables

TL;DR

You'll learn to solve systems where two linear equations share the same variables. There are three methods: graphing, substitution, and elimination. Each system has exactly one solution, no solution, or infinitely many solutions.

1. The Mental Model

A system is just two lines on a coordinate plane. When you solve the system, you're finding where those lines intersect. Sometimes they cross at one point, sometimes they're parallel and never meet, sometimes they're the same line. That's the whole idea.

2. The Core Material

Understanding What You're Looking For

When you have two equations like $2x + 3y = 7$ and $x - y = 1$, you're looking for the values of $x$ and $y$ that make BOTH equations true at the same time. Think of it as finding coordinates $(x, y)$ that satisfy both conditions simultaneously.

Every system has exactly one of three outcomes:
- One solution: The lines intersect at exactly one point
- No solution: The lines are parallel (same slope, different y-intercepts)
- Infinitely many solutions: The lines are identical (same slope, same y-intercept)

Method 1: Solving by Substitution

This is often the cleanest method when one equation is easy to solve for a single variable.

Steps:
1. Solve one equation for one variable
2. Substitute that expression into the other equation
3. Solve for the remaining variable
4. Substitute back to find the first variable

Let's say you have:
$$x + 2y = 8$$
$$3x - y = 1$$

From the first equation: $x = 8 - 2y$

Substitute into the second: $3(8 - 2y) - y = 1$

Simplify: $24 - 6y - y = 1$, so $24 - 7y = 1$, which gives $y = \frac{23}{7}$

Then $x = 8 - 2(\frac{23}{7}) = 8 - \frac{46}{7} = \frac{10}{7}$

Method 2: Solving by Elimination

This method works by adding or subtracting equations to eliminate one variable. You'll often need to multiply one or both equations by constants first.

Steps:
1. Line up the equations with variables aligned
2. Multiply equations by constants to make coefficients of one variable opposites
3. Add the equations to eliminate that variable
4. Solve for the remaining variable
5. Substitute back to find the eliminated variable

Using the same system:
$$x + 2y = 8$$
$$3x - y = 1$$

Multiply the second equation by 2: $6x - 2y = 2$

Now add to the first equation:
$(x + 2y) + (6x - 2y) = 8 + 2$
$7x = 10$
$x = \frac{10}{7}$

Substitute back: $\frac{10}{7} + 2y = 8$, so $y = \frac{23}{7}$

Metho

Linear Inequalities and Applications

TL;DR

Linear inequalities show relationships where one expression is greater than, less than, or equal to another. You'll solve them like equations but flip the inequality sign when multiplying or dividing by negative numbers. They're essential for modeling real-world constraints like budgets, time limits, and resource allocation.

1. The Mental Model

Think of a linear inequality as drawing a line on a graph, then shading one side to show all the solutions that work. Unlike equations that give you one answer, inequalities give you a whole region of possibilities. That's the whole idea.

2. The Core Material

Basic Inequality Solving

You solve linear inequalities almost exactly like linear equations. Add, subtract, multiply, and divide to isolate your variable. There's just one crucial rule: when you multiply or divide both sides by a negative number, flip the inequality sign.

Let's see why this matters:
- Start with: 5 > 3 (true)
- Multiply both sides by -1: -5 > -3 (false!)
- Flip the sign: -5 < -3 (true again)

Here's your step-by-step process:
1. Simplify both sides (distribute, combine like terms)
2. Move variables to one side, constants to the other
3. Divide by the coefficient of your variable
4. If that coefficient is negative, flip the inequality sign

Example: Solve -3x + 7 ≥ 16
- Subtract 7: -3x ≥ 9
- Divide by -3 (flip the sign): x ≤ -3

Graphing Solutions

When you solve x ≤ -3, you're not done yet. You need to understand what this means visually. On a number line, you'd put a solid dot at -3 (because x can equal -3) and shade everything to the left.

The inequality symbols tell you exactly what to draw:
- < or >: Open circle (value not included)
- ≤ or ≥: Solid dot (value included)
- <: Shade left
- >: Shade right

For two-variable inequalities like y < 2x + 1, you'll graph the boundary line y = 2x + 1, then shade the region that satisfies the inequality. Test a point (usually the origin if it's not on the line) to see which side to shade.

Real-World Applications

This is where inequalities become powerful. Most real problems don't have exact answers – they have constraints and ranges of acceptable solutions.

Budget problems: "You have $50 to spend on books costing $12 each and notebooks costing $3 each." If x = books and y = notebooks, then 12x + 3y ≤ 50.

Time constraints: "A project takes 2 hours per task A and 3 hours per task B, with at most 4

Advanced Topics and Review

TL;DR

You'll master systems of linear equations, parametric solutions, and matrix applications. We'll connect everything you've learned about slopes, intercepts, and graphing into a unified framework. This review solidifies your foundation for advanced mathematics.

1. The Mental Model

Linear equations are the building blocks of mathematical relationships. Everything connects: slopes become coefficients, intercepts become constants, and graphs become solution sets. When you have multiple equations, you're finding where these relationships intersect. That's the whole idea.

2. The Core Material

Systems of Linear Equations

When you have multiple linear equations working together, you're dealing with a system. The key insight? You're looking for values that satisfy ALL equations simultaneously.

Consider this system:

2x + 3y = 12
x - y = 1

You have three methods to solve this:

Substitution: Solve one equation for a variable, then substitute into the other. From the second equation: x = y + 1. Substitute into the first: 2(y + 1) + 3y = 12. This gives you 2y + 2 + 3y = 12, so 5y = 10, and y = 2. Then x = 2 + 1 = 3.

Elimination: Multiply equations to eliminate a variable. Multiply the second equation by 3: 3x - 3y = 3. Add to the first equation: 2x + 3y + 3x - 3y = 12 + 3. This gives 5x = 15, so x = 3. Substitute back to get y = 2.

Graphing: Plot both lines and find their intersection point. This works great for visualization but isn't always precise for complex numbers.

Types of Solutions

Systems can behave in three ways. One solution means the lines intersect at exactly one point - they have different slopes. No solution means parallel lines that never meet - same slope, different y-intercepts. Infinite solutions means the equations represent the same line - identical slope and y-intercept.

Here's how to spot each type algebraically. If you eliminate variables and get something like 5 = 5 (always true), you have infinite solutions. If you get 0 = 7 (impossible), there's no solution. If you get x = 3 or y = -2 (specific values), you have exactly one solution.

Parametric Solutions

Sometimes you'll encounter systems with more variables than equations, or equations that aren't independent. This creates parametric solutions - solutions expressed in terms of a parameter.

For example, if you have x + 2y - z = 3, you can't find unique values. Instead, let z = t (a para