# Linear Equations - Definition, Forms, and Graphs

## What Is a Linear Equation?
A **linear equation** is an equation where the variable (or variables) appear only to the first power. The simplest case has one variable:

`ax+b=0`

and has exactly one solution. With two variables — the more common case for graphing — a linear equation looks like:

`Ax+By=C`

and describes a _line_ in the coordinate plane. Every point `(x,y)` that satisfies the equation lies on that line; every point on the line satisfies the equation.

### A few examples:
- `3x+5=14` — one variable, solution `x=3`
- `y=2x+1` — two variables, graph is a line with slope `2` and y-intercept `1`
- `4x−3y=12` — two variables, standard form

The defining feature is _first-power variables only_. The moment you see `x^2`, `sqrt{x}`, `1/x`, or `xy`, the equation is no longer linear.

## The Three Standard Forms
The same line can be written in three different forms — each useful for different purposes. Knowing how to read each form and convert between them is the practical core of linear equations.

### 1. Standard Form
`Ax+By=C`
where `A`, `B`, and `C` are constants (usually written so `A`, `B`, `C` are integers and `A≥0`). Standard form is the most general — it can represent vertical lines (when `B=0`), which the other forms cannot.

**Example.** `4x+3y=12`.  
**Useful for:** Finding the x-intercept and y-intercept quickly. Set `y=0` to find the x-intercept: `4x=12`, so `x=3`. Set `x=0` to find the y-intercept: `3y=12`, so `y=4`. The line goes through `(3,0)` and `(0,4)`.

### 2. Slope-Intercept Form
`y=mx+b`
where `m` is the **slope** (rise over run) and `b` is the **y-intercept** (where the line crosses the y-axis).

**Example.** `y=2x+1`. Slope `m=2` (the line rises 2 units for every 1 unit right); y-intercept `b=1` (the line crosses the y-axis at `(0,1)`).

**Useful for:** Graphing. Plot the y-intercept first, then use the slope to find a second point.

### 3. Point-Slope Form
`y−y1=m(x−x1)`
where `m` is the slope and `(x1,y1)` is a specific point the line passes through.

**Example.** A line through `(2,5)` with slope `3`: `y−5=3(x−2)`.  
**Useful for:** Writing the equation of a line when you know the slope and one point — or when you know two points (compute the slope first, then plug in either point).

## Slope — What It Measures
The **slope** of a line measures its steepness — how much `y` changes when `x` changes by 1 unit. The formula, given two points `(x1,y1)` and `(x2,y2)` on the line:

`m=(y2−y1)/(x2−x1)`  
- **Positive slope** — line goes up from left to right.
- **Negative slope** — line goes down from left to right.
- **Zero slope** — horizontal line (`y = constant`).
- **Undefined slope** — vertical line (`x = constant`).

The slope is the _rate of change_.

## How to Graph a Linear Equation
Three methods cover almost every situation.

### Method 1: From Slope-Intercept Form
For `y=mx+b`:
1. Plot the y-intercept `(0,b)`.
2. From that point, use the slope: rise (numerator) up, run (denominator) right. Plot a second point.
3. Draw a line through the two points.

**Example.** `y=1/2*x−3`
- y-intercept: `(0,−3)`.
- Slope `1/2`: from `(0,−3)`, go up `1` and right `2` to reach `(2,−2)`.
- Draw the line.

### Method 2: From Standard Form (Using Intercepts)
For `Ax+By=C`:
1. Set `y=0` to find the x-intercept: `Ax=C`, so `x=C/A`.
2. Set `x=0` to find the y-intercept: `By=C`, so `y=C/B`.
3. Plot both intercepts. Draw a line through them.

### Method 3: From a Table of Values
Pick three values of `x`, compute the corresponding `y`, plot, draw.

## How Do You Solve a Linear Equation in One Variable?
Solving a linear equation in one variable means isolating that variable on one side using the four legal moves (add, subtract, multiply, divide both sides by the same non-zero quantity).

**The 4-step procedure.**
1. **Clear fractions and parentheses.** Multiply through to remove denominators; distribute to remove parentheses.
2. **Combine like terms** on each side.
3. **Move the variable to one side** and constants to the other (add/subtract).
4. **Divide by the coefficient** of the variable.

## What Are Linear Equations in Two Variables?
A **linear equation in two variables** has the form:

`ax+by+c=0` (or equivalently `ax+by=−c`).

Unlike one-variable equations, a single equation in two variables has _infinitely many_ solutions.

## How Do You Solve a System of Linear Equations?
A **system of linear equations** is a set of two (or more) linear equations to be satisfied simultaneously.

## Methods
### Method 1: Substitution
### Method 2: Elimination
### Method 3: Graphing

## Number of Solutions
| Geometric Picture | Algebraic Sign | Number of Solutions |
| --- | --- | --- |
| Two lines cross at one point | Slopes differ | One unique solution |
| Two lines are the same line | Slopes equal, intercepts equal | Infinitely many |
| Two parallel lines (never meet) | Slopes equal, intercepts differ | None (inconsistent) |

## Where Linear Equations Appear in the Real World
- **Speed and distance.** A car travelling at constant speed traces a linear equation.
- **Currency conversion.** Converting US dollars to Indian rupees uses the exchange rate.
- **Phone plan pricing.** 
- **Temperature conversion.** Fahrenheit to Celsius.
- **Linear regression.** The "line of best fit" through a scatter plot.

## The Mathematicians Who Shaped Linear Equations
**René Descartes** and **Pierre de Fermat** independently developed the concept of analytic geometry
and **Muhammad ibn Musa al-Khwarizmi** systematized procedures for solving linear equations algebraically.

## A Practical Next Step
Try these three problems:
1. Write the equation of the line with slope `-2` and y-intercept `5`.
2. Find the slope of the line through `(2,3)` and `(5,12)`.
3. Convert `4x−2y=8` to slope-intercept form.