Line Forms & Special Cases
See slope-intercept, point-slope, and standard form as different descriptions of the same line, then learn which forms stay useful when the line is horizontal or vertical.
Why would we rewrite a line if the line itself has not changed?
Different forms expose different information. Rewriting is useful when it makes the next question easier, while preserving exactly the same set of ordered-pair solutions.
Equivalent forms are not three nearby lines. They are three algebraic views of one identical geometric object.
One line, three forms.
The line y = 2x − 3 passes through (2, 1). Rewriting it changes which facts are visible first.
Makes slope m and y-intercept b immediately visible.
Choose a form because of the information you already have or need next.
Best when slope and y-intercept are known or when you want to graph from those two features.
Best when you know a slope and any point, without first calculating the y-intercept.
Useful for symmetric algebra, intercepts, systems, and vertical lines such as x = 3.
Rewrite the line without changing it.
Horizontal and vertical lines expose the limits of the forms.
Slope is 0. Slope-intercept form works perfectly: m = 0 and b = −2.
Slope is undefined. Slope-intercept and point-slope require a finite slope, but standard form handles x = 3 directly.