## Introduction

A **linear equation** is the **equation of a line**, and the **graphs of linear equations** always display a **straight line**. All linear equations **must include variables** such as x and y and **may include a constant or coefficient** to a variable but there will be **no exponent**.

Here is an example of a linear equation and some points on the line $y=2x +4$. A chart is a handy tool you can use to create a list of some of the **ordered pairs** of an equation. Because the equation is linear, the graph of this set of points is a straight line.

$y=2x +4$

$\begin{array}{|lrl|} \hline&&\\[-3mm] & \mathbf{x}& \mathbf{y}\\[0.5mm] \hline&&\\[-3mm] & -2 & 0\\[0.5mm] & -1 & 2\\[0.5mm] & 0 & 4\\[0.5mm] & 1 & 6\\[0.5mm] & 2 & 8\\[0.5mm] \hline \end{array}$

## Slope

The **slope** of a line is the **slant or incline of a line**. Slope can be a **positive or negative number**.The symbol **m** is used to represent the slope. Use the **slope formula** to calculate the value of the slope equal to the change of y divided by the change of x.

$m=\frac{y_2-y_1}{x_2-x_1}$

**Example:**
Find the slope of the line that passes through points (2, 11) and (3, 9).

$m=\frac{9-11}{3-2} = \frac{-2}{1}= -2$

The slope is equal to -2. The negative sign indicates a downward slope.

## Writing Linear Equations

Although there are **several formulas** for writing equations of lines, linear equations are most often presented in the **Slope-Intercept form**.

### Slope-Intercept Form

$y=mx+b$

- m is the slope of the line
- b is the y-intercept

### Standard Form

The **standard form** of a line is another format to present the same information as the slope-intercept form.

$Ax+By=C$

A, B, and C are coefficients or constants.

To change from standard form to slope-intercept form, use **opposite operations** to move the terms.

$\begin{align} Ax+By&=C\\ -3x + y &= 16\\ -3x +3x + y &= 16 +3x\\ y &= 3x +16 \end{align}$

$\begin{align} y&=mx+b\\ y &= 3x +16 \end{align}$

Writing the equation in the slope-intercept form makes it easier to determine the slope of this line is 3, and the y-intercept is 16.

### Point-Slope Form

The **point-slope form** is used to calculate an unknown point when the slope and one point on the line is known.

$ (y-y_1)=m(x-x_1)$

y and x are unknown values. $y_1$ and $x_1$ are a known ordered pair.

Use the point-slope form to write the equation of the line with a slope of 1 and passing through point (3, 4).

$\begin{align} (y-y_{1})&=m(x-x_{1})\\ (y-4)&=1(x-3)\\ y -4 &= x -3\\ y - 4 + 4 &= x -3 +4\\ y&=x + 1 \end{align}$

## Graphing Linear Equations

Use the information found in the equation of a line written in slope-intercept form to create the matching **graph**.

$\begin{align} y&=mx+b\\ y &= 2x +3 \end{align}$

For this linear equation, the slope is 2 and the y-intercept is 3.

## Parallel and Perpendicular Lines

The slope can be used to indicate if lines are **parallel** or **perpendicular**:

**Parallel lines: slopes are equal****Perpendicular lines: slopes are negative inverses**

Example: For this problem, are the slopes parallel or perpendicular? Compare the equation of two lines. What do you know about the slopes?

$\begin{align} y&=2x +4\\ y&=2x -5 \end{align}$

The slopes are the same, so the lines are parallel.

$\begin{align} y&= -2x+1\\ y&=\frac{1}{2}x+3\\ \end{align}$

The slopes are negative inverses (the product of the two slopes is equal to -1), so the lines are perpendicular.

## Direct Variation

When the value of y varies with the value of x, there is a **direct variation**. The formula for direct variation is:

$\begin{align} y&= k\times x\\ y&= kx \end{align}$

k is the **constant of variation**. The formula for k is:
$k=\frac{y}{x}$

**Example:**
This problems demonstrates a real world use of direct variation. A car gets 20 miles per gallon. How many gallons of gas are needed to travel a distance of 2,400 miles? Use the direct variation to solve for x.

$\begin{align} y&= kx\\ y&=20x\\ 2,\!400 &= 20x\\ \frac{2,\!400}{20} &= \frac{20}{20}x\\ x&=120 \end{align}$

With a constant of variation of 20, the car will need 120 gallons to travel 2,400 miles.

## Graphing Linear Inequalities

Graphing linear inequalities is a multi-step process. First graph the line. Then determine if the graph includes the line and if the solution set is above or below the line.

**Example 1:**
For this inequality, the line is not included, and the solution set is below the line.
$y<3x +4$

**Example 2:**
But for this inequality, the line is included, and the solution set is above the line.
$y\geq\frac{1}{2}x +5$.