The equation of a line describes every point that lies on a straight path in the coordinate plane. It is one of the most useful tools in algebra and geometry because it connects graphs, tables, and real-world rates of change. Once you know a line's slope and at least one point, you can write an equation that represents the entire line.
This makes linear equations powerful for modeling motion, cost, distance, temperature, and many other patterns.
The slope tells how steep the line is and whether it rises or falls from left to right. The y-intercept tells where the line crosses the y-axis, which often represents a starting value. Slope-intercept form is best for graphing quickly, while point-slope form is best when you are given a point and a slope.
When you are given two points, first find the slope, then substitute one point into a line equation form.
Key Facts
- Slope formula: m = (y2 - y1)/(x2 - x1)
- Slope-intercept form: y = mx + b
- Point-slope form: y - y1 = m(x - x1)
- A positive slope rises from left to right, and a negative slope falls from left to right.
- The y-intercept is the point where x = 0, written as (0, b).
- Parallel nonvertical lines have the same slope, and perpendicular nonvertical lines have slopes whose product is -1.
Vocabulary
- Slope
- Slope is the ratio of vertical change to horizontal change between two points on a line.
- Y-intercept
- The y-intercept is the point where a line crosses the y-axis.
- Slope-intercept form
- Slope-intercept form is the equation y = mx + b, where m is the slope and b is the y-intercept.
- Point-slope form
- Point-slope form is the equation y - y1 = m(x - x1), which uses a known point and the slope.
- Linear equation
- A linear equation is an equation whose graph is a straight line.
Common Mistakes to Avoid
- Swapping rise and run, which gives the reciprocal of the correct slope. Always compute vertical change over horizontal change, so m = change in y divided by change in x.
- Forgetting the sign of the slope, which changes the direction of the line. If the line falls from left to right, the slope must be negative.
- Using the x-intercept as b in y = mx + b, which places the line incorrectly. The value b is the y-value where the line crosses the y-axis.
- Substituting a point incorrectly into point-slope form, which creates an equation for the wrong line. Match x1 with the point's x-coordinate and y1 with the point's y-coordinate.
Practice Questions
- 1 Find the equation of the line in slope-intercept form that passes through (2, 5) and (6, 13).
- 2 A line has slope -3 and passes through the point (4, 1). Write the equation in point-slope form, then convert it to slope-intercept form.
- 3 Two students graph the line y = 2x - 3. One starts at (0, -3) and moves up 2 and right 1. The other starts at (0, -3) and moves down 2 and left 1. Explain why both methods produce points on the same line.