Understanding Linear Equations and Their Graphs

This page introduces two key forms of linear equations: slope-intercept form and point-slope form. These forms are crucial for graphing linear equations step by step.

Slope-Intercept Form

The slope-intercept form is expressed as y = mx + b, where:

• m represents the slope
• b represents the y-intercept

Example: y + 2 = 2/3(x - 3)

This form is particularly useful when you need to quickly identify the slope and y-intercept of a line.

Point-Slope Form

The point-slope form is expressed as y₂ - y₁ = m(x₂ - x₁), where:

• (x₁, y₁) is a known point on the line
• m is the slope

Example: Given a point (-1, 3) and slope -4, the equation would be: y - 3 = -4(x + 1)

This form is helpful when you're given a point on the line and its slope.

Graphing a Line

To graph a line using the point-slope form:

1. Plot the given point
2. Use the slope to mark another point

Example: For the equation y + 3 = 2(x - 4), which passes through (4, -3) with a slope of 2:

1. Plot the point (4, -3)
2. Use the slope to find another point: moving right 1 unit and up 2 units

Highlight: Understanding these forms is crucial for graphing linear equations in two variables efficiently.

Advanced Concepts in Linear Equations

This page delves into the standard form of linear equations, finding intercepts, and special cases of horizontal and vertical lines.

Standard Form

The standard form of a linear equation is ax + by = c, where a, b, and c are constants and a and b are not both zero.

Example: To convert y - 1 = 3(x - 1) to standard form:

1. Expand: y - 1 = 3x - 3
2. Rearrange: 3x - y = -2

Finding Intercepts

Intercepts are crucial points where a line crosses the x or y axis.

To find the x-intercept:

• Set y = 0 and solve for x

Example: For 3x + 4y = 8 3x + 4(0) = 8 x = 8/3

To find the y-intercept:

• Set x = 0 and solve for y

Example: For 3x + 4y = 8 3(0) + 4y = 8 y = 2

Horizontal and Vertical Lines

Horizontal lines have the form y = a, where a is a constant. Vertical lines have the form x = b, where b is a constant.

Highlight: Vertical lines can only be written in standard form.

Vocabulary: A linear equation graph solver can be a helpful tool for visualizing these concepts.

Understanding these advanced concepts allows for a comprehensive grasp of linear equations and their graphical representations, essential for solving complex problems in algebra and geometry.