Finding Slope

2 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

You will meet a line described in several different ways. It might be given as two points, as an equation, or as an angle.

Each one leads to the same slope. You just need the right method for the form you are given.

Rule — From two points

For distinct points $A(x_1,y_1)$ and $B(x_2,y_2)$ ,

m = \dfrac{y_2-y_1}{x_2-x_1}

Subtract in the same order upstairs and downstairs. If $x_2-x_1=0$ , the line is vertical and its slope is undefined.

Worked example — Two points

Find the slope through $A(-4,2)$ and $B(3,5)$ . The rise is $5-2=3$ , while the run is $3-(-4)=7$ .

m = \dfrac{5-2}{3-(-4)} = \dfrac{3}{7}

Rule — From an equation

Rewrite the equation in slope-intercept form $y=mx+b$ . The coefficient $m$ is the slope. For standard form $ax+by+c=0$ with $b\ne0$ ,

m = -\dfrac{a}{b}

A horizontal equation $y=k$ has slope $0$ ; a vertical equation $x=k$ has undefined slope.

Worked example — An equation

Find the slope of $3y=4x-6$ . Divide every term by $3$ to isolate $y$ :

y = \dfrac{4}{3}x-2

The coefficient of $x$ is the slope, so $m=\dfrac{4}{3}$ .

Rule — From an inclination angle

If a line makes an angle $\alpha$ with the positive $x$ -axis, measured counterclockwise, then

m=\tan(\alpha)

Acute angles give positive slopes, obtuse angles give negative slopes, $0^\circ$ gives slope zero, and $90^\circ$ gives an undefined slope.

Explore

Drag the two points. The readout gives the rise, the run, and the slope as a reduced fraction. Now put both points at the same height. The rise becomes zero, so the slope is zero. Put them one above the other and the run becomes zero, so the slope is undefined.

Rise over run

Try this. Give A and B the same y for zero slope, then the same x for undefined slope. Coincident points do not determine a slope.

Point A · x-2Point A · y3Point B · x3Point B · y1

A = (-2, 3), B = (3, 1) · Rise = -2 · Run = 5 · Slope = -0.4

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

Find the slope through

(1,5)

and

(3,1)

$-1/2$

$-2$

$2$

Hint 1 · Find a starting point

Use the same point order in numerator and denominator.

Hint 2 · Take the next step

Compute (1 − 5)/(3 − 1).

Show the reasoning

Answer: $-2$

The slope is $-4/2=-2$ . As x increases, y decreases, so the sign also agrees with the graph.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

Optional marks, not a grade. Saved in this browser only. Open notebook →