Linear equations describe anything that changes at a steady rate: a phone plan that costs a fixed amount plus a fee per gigabyte, a car moving at constant speed, savings that grow by the same amount each week. Master slope and the three forms of a line and a large part of algebra falls into place.
What slope means
Slope measures steepness and direction: how much y changes for each one-unit change in x. Between two points (x₁, y₁) and (x₂, y₂):
m = (y₂ − y₁) / (x₂ − x₁), often remembered as "rise over run."
- Positive slope: the line rises from left to right.
- Negative slope: it falls from left to right.
- Zero slope: a horizontal line, like y = 4.
- Undefined slope: a vertical line, like x = 2, because the run is zero and you cannot divide by zero.
In real problems, slope is a rate. If a line shows cost against hours with slope 15, it means the cost rises by 15 dollars per hour.
Worked example 1: slope from two points
Find the slope through (2, 3) and (6, 11).
m = (11 − 3) / (6 − 2) = 8 / 4 = 2. For every step right, the line goes up 2.
Form 1: slope-intercept form
y = mx + b, where m is the slope and b is the y-intercept, the point where the line crosses the y-axis (0, b). This is the easiest form to graph: plot (0, b), then use the slope to find more points.
Example: y = −½x + 4 crosses the y-axis at 4. From (0, 4), go down 1 and right 2 to reach (2, 3), and keep going.
Form 2: point-slope form
y − y₁ = m(x − x₁). Use it when you know a point and the slope, or two points.
Worked example 2: equation through two points
Write the equation of the line through (1, 5) and (3, 9).
- Slope: m = (9 − 5) / (3 − 1) = 2.
- Point-slope with (1, 5): y − 5 = 2(x − 1).
- Simplify: y − 5 = 2x − 2, so y = 2x + 3.
- Check with the other point: 2(3) + 3 = 9. Correct.
Form 3: standard form
Ax + By = C, where A, B and C are integers and A is usually positive. It makes intercepts easy: set y = 0 to find the x-intercept and x = 0 to find the y-intercept. For 3x + 2y = 12, the x-intercept is (4, 0) and the y-intercept is (0, 6).
Parallel and perpendicular lines
- Parallel lines have equal slopes and never meet.
- Perpendicular lines have slopes that are negative reciprocals; they multiply to −1. A line with slope 2/3 is perpendicular to one with slope −3/2.
Worked example 3: perpendicular line through a point
Find the line perpendicular to y = 4x − 1 that passes through (8, 2).
- The original slope is 4, so the perpendicular slope is −¼.
- y − 2 = −¼(x − 8), so y − 2 = −¼x + 2 and y = −¼x + 4.
Common mistakes
- Subtracting in a different order on top and bottom. If you use y₂ − y₁, you must use x₂ − x₁.
- Mixing up the intercept: in y = mx + b, b is the y-intercept, not the x-intercept.
- Saying a vertical line has zero slope. Its slope is undefined.
- Forgetting to distribute the slope in point-slope form.
Practice questions
- Find the slope through (−2, 4) and (3, −6).
- Write y − 3 = 5(x + 1) in slope-intercept form.
- Find the intercepts of 2x − 5y = 10.
- Are y = 3x + 2 and y = −⅓x − 7 parallel, perpendicular or neither?
Answers: (1) m = (−6 − 4) / (3 + 2) = −2. (2) y = 5x + 8. (3) x-intercept (5, 0), y-intercept (0, −2). (4) Perpendicular, because 3 × (−⅓) = −1.
Last reviewed 2026-10-23 by the BrainBoost team.