doing math

courses › Algebra

Lines & slope

level 21 course

Slope, intercepts, and perpendicular lines.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 21.

Sign in to start

Builds on: Linear equations (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

A straight line has one steady steepness, its slope: the change in y divided by the change in x, or rise over run.

Most lines can be written y = mx + b, where m is the slope and b is the y-intercept, the height where the line crosses the y-axis. The x-intercept is where it crosses the x-axis, so y is 0 there.

Perpendicular lines, which meet at a right angle, have slopes that are negative reciprocals of each other.

Techniques

Rise over run

You know two points on the line.

  1. Rise: the second y-value minus the first.
  2. Run: the second x-value minus the first, in the same order.
  3. Divide the rise by the run, keep the sign, and reduce the fraction.
worked example

Example: What is the slope of the line through (1, −2) and (4, 7)?

  1. Rise: 7 − (−2) = 9.
  2. Run: 4 − 1 = 3.
  3. Slope: 9 ÷ 3 = 3.

Answer: 3

Put the point in

Finding b from the slope and a point, or finding where the line meets the x-axis.

  1. For b: write y = mx + b with the slope in place of m.
  2. Put in the point, x and y, and solve for b.
  3. For the x-intercept: set y to 0 and solve for x. Leave a fraction in lowest terms.
worked example

Example: A line has slope 2 and passes through (3, 1). What is its y-intercept?

  1. Write y = 2x + b and put in the point: 1 = 2 × 3 + b.
  2. So 1 = 6 + b, and b = 1 − 6 = −5.

Answer: −5

Flip and change the sign

You want the slope of a line perpendicular to a given one.

  1. Flip the slope upside down: 2/3 becomes 3/2, and 4 becomes 1/4.
  2. Change its sign: positive becomes negative, negative becomes positive.
  3. Check: the two slopes multiply to −1.
worked example

Example: A line has slope −4/3. What is the slope of a line perpendicular to it?

  1. Flip −4/3 to −3/4.
  2. Change the sign: 3/4.
  3. Check: −4/3 × 3/4 = −1.

Answer: 3/4

Tips by skill

  • TipSlope from two points: Rise over run: subtract the y-values, then the x-values in the same order, and divide. Reduce the fraction.
  • Tipy-intercept: Put the point into y = mx + b and solve for b. Watch the sign when you move m times x across.
  • Tipx-intercept: The line meets the x-axis where y is 0. Set y to 0, solve for x, and keep a fraction in lowest terms.
  • TipPerpendicular slope: Flip the slope upside down and change its sign. The two slopes should multiply to −1.

Watch out for

  • Dividing the run by the rise. Slope puts the change in y on top.
  • Subtracting in one order on top and the other order on the bottom. Start from the same point in both.
  • Flipping a slope for a perpendicular line but keeping its sign.
  • Giving the y-intercept when the question asks where the line crosses the x-axis.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

your rounds

No rounds yet.