Graphing Linear Equations: Slope, Intercepts, and Forms

Section 2.3b: Graphing Linear Equations

Slope-Intercept Form

The slope-intercept form of a linear equation is a fundamental way to represent lines in algebra. It is written as:

• m represents the slope of the line.

• b represents the y-intercept, the point where the line crosses the y-axis.

The slope-intercept form is:

$y = mx + b$

To graph a line using this form:

1. Plot the y-intercept (b) on the y-axis.

2. From the y-intercept, use the slope (m) to determine the "rise" (change in y) and "run" (change in x) to locate a second point.

3. Draw a straight line through the two points.

Example: For $y = 2x + 3$, the slope is 2 and the y-intercept is 3. Plot (0,3), then from there move up 2 units and right 1 unit to plot the second point.

Equations of Horizontal and Vertical Lines

• Horizontal lines have a slope of zero. Their equation is:

• $y = b$

• Vertical lines have an undefined slope. Their equation is:

• $x = a$

Example: $y = 4$ is a horizontal line passing through y = 4. $x = -2$ is a vertical line passing through x = -2.

General Form of a Linear Equation

The general form of a linear equation is:

$Ax + By + C = 0$

where A and B are not both zero.

To convert to slope-intercept form:

• Solve for y to get $y = mx + b$.

Example: Rewrite $2x - 3y + 6 = 0$ in slope-intercept form:

$2x - 3y + 6 = 0$ $-3y = -2x - 6$ $y = \frac{2}{3}x + 2$

Using Intercepts to Graph $Ax + By + C = 0$

Another method to graph a line is by finding its intercepts:

1. x-intercept: Set $y = 0$ and solve for $x$.

2. y-intercept: Set $x = 0$ and solve for $y$.

3. Plot both intercepts and draw a line through them.

Example: For $3x + 2y - 6 = 0$:

• x-intercept: $y = 0 \Rightarrow 3x - 6 = 0 \Rightarrow x = 2$

• y-intercept: $x = 0 \Rightarrow 2y - 6 = 0 \Rightarrow y = 3$

Plot (2,0) and (0,3), then draw the line.

Summary of Equations of Lines

Form	Equation	Description
Point-Slope Form	$y - y_1 = m(x - x_1)$	Line with slope m passing through $(x_1, y_1)$
Slope-Intercept Form	$y = mx + b$	Line with slope m and y-intercept b
Horizontal Line	$y = b$	Slope = 0
Vertical Line	$x = a$	Slope is undefined
General Form	$Ax + By + C = 0$	A and B not both zero

Additional info:

• Graphing linear equations is a foundational skill in algebra, used to model relationships and solve real-world problems.

• Understanding the different forms of linear equations helps in identifying key features such as slope and intercepts quickly.