LEMMA

Lines & systems

Slope from two points

ALEKS placement

Slope is rise over run: the change in y divided by the change in x.

What this covers

m=x2​−x1​y2​−y1​​
  • Sketch the two points and connect them. Slope compares how far the line climbs to how far it travels sideways.
  • Pick one point to go first and keep that order: its y first on top, its x first on the bottom.
  • A negative difference is a direction, not a mistake. Keep it — a line that falls left to right has a negative slope.
  • Check by walking the points the other way: both differences flip sign, so the slope comes out the same.
  • Reduce the fraction. Zero on top is a horizontal line; zero on the bottom is undefined, a vertical line.

Worked example

Worked example

Find the slope through (1, -2) and (5, 6).

  1. m=5−16−(−2)​Second point minus first, in both coordinates.
  2. m=48​=2Rise 8 over run 4 reduces to 2.
Another worked example

Find the slope through (−1, 2) and (3, −4).

  1. m=3−(−1)−4−2​Subtract the first point's coordinates from the second point's in both places. The rise is negative six and the run is four.
  2. m=−23​Negative six over four reduces to negative three halves, so the line falls three units for every two units to the right.

A common mistake

A mistake Lemma catches

From

m=1−56−(−2)​m=5−16−(−2)​

“The top takes the second point minus the first, 6 − (−2), but the bottom takes the first minus the second, 1 − 5. Mixing the orders flips the sign: 8/(−4) = −2. Keep one order on both: (6 − (−2))/(5 − 1) = 8/4 = 2.”

Try one

Sample problem

Find the slope through (-2, -5) and (-1, -2).

(−2,−5),(−1,−2)

Practice slope free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

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