Click image to open full size
Linear Equations: Slope & Intercepts
Slope-Intercept Form, Standard Form, and Graphing
Related Tools
Related Labs
Linear equations describe relationships that change at a constant rate, and their graphs are straight lines. Two of the most important features of a line are its slope and its intercepts. Slope tells how steep the line is and whether it rises or falls, while intercepts show where the line crosses the axes. These ideas are used in algebra, physics, economics, and any situation involving steady change.
A line is often written in slope intercept form as y = mx + b, where m is the slope and b is the y-intercept. The slope can be found from two points using m = (y2 - y1)/(x2 - x1), and the x-intercept is found by setting y = 0. On a graph, a slope triangle helps show rise over run visually, making the rate of change easier to interpret. Understanding how these pieces connect lets students move between equations, tables, and graphs with confidence.
Key Facts
- Slope measures rate of change: m = rise/run = (y2 - y1)/(x2 - x1)
- Slope intercept form is y = mx + b, where m is slope and b is the y-intercept
- The y-intercept is the point where x = 0, so its coordinates are (0, b)
- The x-intercept is the point where y = 0, found by solving 0 = mx + b
- A positive slope means the line rises from left to right, and a negative slope means it falls
- Horizontal lines have slope 0, while vertical lines have undefined slope
Vocabulary
- slope
- Slope is the ratio of vertical change to horizontal change and describes the steepness of a line.
- y-intercept
- The y-intercept is the point where a graph crosses the y-axis.
- x-intercept
- The x-intercept is the point where a graph crosses the x-axis.
- rise over run
- Rise over run is a visual way to compute slope by comparing vertical change to horizontal change.
- linear equation
- A linear equation is an equation whose graph is a straight line and whose variables are raised only to the first power.
Common Mistakes to Avoid
- Switching the order of subtraction in the slope formula, because if you subtract x-values in one order and y-values in the opposite order you can get the wrong sign for the slope.
- Confusing the y-intercept with the x-intercept, because the y-intercept happens when x = 0 and the x-intercept happens when y = 0.
- Reading slope as run over rise, because slope is defined as rise divided by run and reversing it changes the value.
- Assuming every equation is already in slope intercept form, because equations often need to be rearranged before the slope and intercept can be identified correctly.
Practice Questions
- 1 Find the slope of the line through the points (2, 5) and (6, 13).
- 2 For the equation y = -3x + 12, find the slope, the y-intercept, and the x-intercept.
- 3 A line has a positive slope and crosses the y-axis below the origin. Describe how the line looks on the coordinate plane and explain what each feature tells you.