Linear Equations: Slopes and Intercepts

Understanding Slopes and Intercepts

In this lesson, we will explore the concepts of slope and intercepts in linear equations. A linear equation in the slope-intercept form is given by:

$y = mx + b$

where:

• $m$ is the slope of the line.
• $b$ is the y-intercept, the point where the line crosses the y-axis.

What is Slope?

The slope ($m$) represents the rate of change of $y$ with respect to $x$. It indicates how steep the line is. The slope can be calculated using the formula:

$m = \frac{\Delta y}{\Delta x} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Finding the Y-Intercept

The y-intercept ($b$) can be found by setting $x = 0$ in the linear equation, which gives the value of $y$ at that point.

Example Problem

Let's find the slope and y-intercept of the line represented by the equation:

$y = 2x + 3$

1. Identify the slope and intercept:
   • Slope ($m$): From the equation, $m = 2$.
   • Y-intercept ($b$): Here, $b = 3$. Thus, the line crosses the y-axis at (0, 3).

Key Questions

• What does the slope tell us about the relationship between $x$ and $y$?
• How can we determine the y-intercept from a linear equation?
• If the slope is negative, what does that indicate about the line?

Summary

To summarize:

• The slope indicates the steepness of the line.
• The y-intercept is where the line crosses the y-axis.

Understanding these concepts is crucial for graphing linear equations and analyzing their behavior.