Math · College algebra · Concept
Slope and equations of lines
The slope of a line measures how steep it is: m = (y₂ − y₁)/(x₂ − x₁), the change in y divided by the change in x between any two of its points. A line with slope m and y-intercept b has the equation y = mx + b, and the line through (x₁, y₁) with slope m is y − y₁ = m(x − x₁). Parallel lines have equal slopes, and perpendicular lines have slopes whose product is −1.
Slope is rise over run
Between two points on a line, the slope is the change in y divided by the change in x. Any two points give the same slope, which is what makes the graph straight. A positive slope rises from left to right, a negative slope falls, a zero slope is horizontal, and a vertical line has no slope because its run is zero.
Slope-intercept form
y = mx + b has slope m and crosses the y-axis at (0, b). It is the quickest form to graph: plot the intercept, then move 1 unit right and m units up, or down if m is negative.
Point-slope form
The line through the point (x₁, y₁) with slope m is y − y₁ = m(x − x₁), or y = y₁ + m(x − x₁). Use it when the known point is not the intercept, then simplify to y = mx + b.
Standard form and special lines
Ax + By = C is standard form, and solving it for y shows the slope −A/B. A horizontal line is y = k, with slope 0. A vertical line is x = h: it is not a function of x, and it has no slope.
Parallel and perpendicular lines
Parallel lines have equal slopes, so distinct parallel lines never meet. Perpendicular lines have negative reciprocal slopes, such as 2/3 and −3/2, whose product is −1. A horizontal line and a vertical line are also perpendicular, even though the vertical one has no slope.
Slope as a rate of change
In a model, slope carries units: output units per input unit. A taxi fare that rises $1.75 per kilometer has slope 1.75 dollars per kilometer, and its intercept is the fare for a 0 km trip.
Common mistakes
- Subtracting coordinates in different orders on the top and bottom: use the same point first in both.
- Reading b off an equation that is not solved for y: 2x − 3y = 6 does not have y-intercept 6.
- Negating a slope for a perpendicular line but forgetting the reciprocal: perpendicular to 2/3 is −3/2, not −2/3.
- Giving a vertical line slope 0: its run is 0, so its slope is undefined.
Key terms
- Slope of a line
- The ratio m = (y₂ − y₁)/(x₂ − x₁) of the change in y to the change in x between two points of a nonvertical line. Every pair of points on the line gives the same slope.
- Slope-intercept form
- The equation y = mx + b of a nonvertical line, where m is the slope and (0, b) is the y-intercept.
- Point-slope form
- The equation y − y₁ = m(x − x₁) of the line with slope m through the point (x₁, y₁).
- Parallel and perpendicular lines
- Distinct nonvertical lines are parallel when their slopes are equal and perpendicular when their slopes multiply to −1. Vertical lines are parallel to one another and perpendicular to horizontal lines.
- x-intercept
- A point where a graph crosses or touches the x-axis, so its y-value is 0. For y = f(x), its x-value is a zero (root) of f.
- y-intercept
- The point where a graph crosses the y-axis, found by setting x = 0: (0, f(0)) when 0 is in the domain.
- Average rate of change
- The change in output divided by the change in input between two distinct points. Geometrically it is the slope of the secant line.
Work through an example
Find the equation of the line through (−2, 7) and (4, −2) in slope-intercept form.
Find the equation of a line through two points →Sources and scope
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Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Graph the line Check the line in Math Open worked example on a board Slope formula in Math ReferenceYour existing work stays on this device. Examples open as editable copies.