Linear Equations Study Guide

Mathematical Foundation of Slope

Slope tells us how steep a line is, just like a hill! We use the letter $m$ for slope.

• Types of Slope:

  • Positive Slope ($m > 0$): Going uphill from left to right!

  • Negative Slope ($m < 0$): Going downhill from left to right!

  • Zero Slope ($m = 0$): Walking on flat ground.

  • Undefined Slope: A straight vertical line, like a wall!

The Slope Formula:
$\text{Slope } (m) = \frac{\text{Rise}}{\text{Run}}$

• Rise: How many steps you move up or down.

• Run: How many steps you move left or right.

Slope-Intercept Form of Linear Equations

This is a simple rule to write and draw a straight line:

$y = m \times x + b$

• $x$ is the input.

• $y$ is the output.

• $m$ is the steepness (slope).

• $b$ is the starting dot on the vertical line, at $(0, b)$.

How to Draw the Line:

1. Put a starting dot at $(0, b)$.

2. Count up or down by the rise, then right or left by the run to place a second dot.

3. Draw a straight line connecting the two dots!

Point-Slope Form of Linear Equations

Use this formula when you know one dot $(x<em>1, y</em>1)$ on the line and how steep it is $m$:

$y - y<em>1 = m \times (x - x</em>1)$

How to Use It:

1. Put the numbers from your dot $(x<em>1, y</em>1)$ into the formula.

2. Put the steepness $m$ into the formula.

3. Multiply and simplify to turn it into $y = m \times x + b$!