The slope of a line measures how steep the line is and which way it leans. Once the slope of a line is known, the slopes of every line parallel or perpendicular to it are known too — and that is what makes it possible to write the equation of a new line through a given point. This page works through all three skills: finding slope, reading slope from an equation, and writing the equation of a parallel or perpendicular line.
Slope. The slope of a nonvertical line is the ratio of the vertical change (the rise) to the horizontal change (the run) between any two points on the line. If (x1, y1) and (x2, y2) are two points on the line, then
m = riserun = y2 − y1x2 − x1
A line that rises from left to right has a positive slope, and a line that falls from left to right has a negative slope. A horizontal line has slope 0 (the rise is 0), and a vertical line has undefined slope (the run is 0, and division by 0 is undefined).
Slope-intercept form. Every nonvertical line can be written as y = mx + b, where m is the slope of the line and b is the y-intercept — the y-coordinate of the point where the line intersects the y-axis. If an equation is not in this form, it must be rewritten in this form before the slope can be read from it: solve the equation for y, and the coefficient of x is the slope.
Parallel lines. Parallel lines are lines that lie in the same plane and do not intersect, no matter how far they are extended. Two or more lines are parallel exactly when they have equal slopes. (Two vertical lines are parallel as well; both have undefined slope.)
Perpendicular lines. Perpendicular lines are lines that intersect at a right angle (90°). Two nonvertical lines are perpendicular exactly when their slopes are opposite reciprocals: flip the fraction and change the sign. Equivalently, the product of the two slopes is −1. A horizontal line and a vertical line are also perpendicular to each other.
What you are looking at: a coordinate grid with a line through two points, A and B. The rise (the vertical change from A to B) and the run (the horizontal change from A to B) are shown under the grid, along with the slope they produce.
What to do: drag point A or point B to a new grid point. The line, the rise, the run, and the slope all update as you drag. Try to make the line rise, fall, lie flat, and stand straight up, and watch what happens to the slope in each case.
What to notice: the slope is the same no matter which two points of the line are used, because the rise and run grow or shrink in the same ratio. A flat line has slope 0, and a vertical line has undefined slope — its run is 0.
What you are looking at: three ways a line can be given — already in slope-intercept form, in standard form, and as two points. Choose one, and its graph appears.
What to do: press Next Step to work through the solution one step at a time. Each step rewrites the given information a little further until the slope can be named. The rise and run that make up the slope are drawn on the graph as the two legs of a right triangle, so the slope can be seen as well as computed.
What to notice: a standard-form equation such as 2x + 4y = 12 does not show its slope directly. The number in front of x is not the slope — the equation must be solved for y first.
What you are looking at: two lines on one grid — a blue line through points A and B, and a red line through points C and D. Each line's slope is written on the graph next to its line, and the box under the grid compares the two slopes.
What to do: drag any of the four points. Move the red line until the verdict under the grid says the lines are parallel, then keep dragging until it says they are perpendicular.
What to notice: the verdict depends only on the two slopes. Equal slopes mean parallel. Slopes whose product is −1 (opposite reciprocals) mean perpendicular. Anything else means the lines intersect, but are neither parallel nor perpendicular.
What you are looking at: a reference line (blue) with its equation, and a point P. Your task, stated above the grid, is to draw the line through P that is either parallel or perpendicular to the reference line — then to write that line's equation.
What to do: drag the orange handle. The red line always stays pinned through P and turns as you drag. When the red line has exactly the slope the task requires, it turns green, its equation appears on the graph, and the box below the grid shows both equations — the reference line's equation and your line's equation — together with the reason the line is correct: the slope of the reference line, the new slope, and the substitution that finds b. The line stays free to move, so keep dragging to explore other slopes whenever you like. Press New Challenge for a new reference line, a new point, and a new task.
What to notice: the new equation is built from just two ingredients — the new slope and the point P. Every parallel or perpendicular writing problem, no matter how the reference line is given, works the same three-step way.
Six questions, one at a time. A first wrong answer earns a hint and one retry; the full explanation appears when a question is answered correctly or after a second miss. Your score counts first-try answers only.