FoundationsConcept reference

Straight-line slope

The slope between two points with different horizontal coordinates is their vertical-coordinate difference divided by their horizontal-coordinate difference.

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Overview
From A=(1,1) to B=(3,4), x increases by 2 while y increases by 3
For A=(1,1) and B=(3,4), reversing both subtractions gives (1-4)/(1-3)=1.5 again
Check your understanding
Key takeaway
Sources & further reading
Concept connections

01From A=(1,1) to B=(3,4), x increases by 2 while y increases by 3#

From $A=(1,1)$ to $B=(3,4)$ , x increases by 2 while y increases by 3. Slope asks how much vertical change corresponds to one horizontal unit.

Figure 1Points A=(1,1) and B=(3,4) form a rise of 3 over a run of 2. Labeled ticks mark units on each axis.

For A=(1,1) and B=(3,4), divide the changes.

m=\frac{4-1}{3-1}=\frac{3}{2}

The slope is $3/2=1.5$ . For different x coordinates, use $m=(y_2-y_1)/(x_2-x_1)$ . Keep the point order the same in both subtractions.

Check your reasoning

Find the slope from $(1,2)$ to $(3,8)$ .

Show answer and explanation

The vertical change is $8-2=6$ and horizontal change is $3-1=2$ . Their ratio is 3.

02For A=(1,1) and B=(3,4), reversing both subtractions gives (1-4)/(1-3)=1.5 again#

For $A=(1,1)$ and $B=(3,4)$ , reversing both subtractions gives $(1-4)/(1-3)=1.5$ again. A horizontal line has slope 0. A vertical line has zero horizontal change, so its slope is undefined.

Check your reasoning

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

A $0$
B $4$
CUndefined

Show answer and explanation

Undefined

Horizontal change is $2-2=0$ , so slope is undefined.

03Check your understanding#

Check your reasoning

A straight-line tank graph goes from 4 liters at minute 1 to 10 liters at minute 4. What is its slope in liters per minute?

Show answer and explanation

The volume changes by 6 liters over 3 minutes: $6/3=2$ .

Key takeaway

Subtract in the same point order, then divide vertical change by horizontal change.

Compute the slope between two distinct points

Sources & further reading

[1]
OpenStax College Algebra 2e: 4 1 Linear Functions ↗OpenStax · Book

01From A=(1,1) to B=(3,4), x increases by 2 while y increases by 3#

02For A=(1,1) and B=(3,4), reversing both subtractions gives (1-4)/(1-3)=1.5 again#

03Check your understanding#

Subtract in the same point order, then divide vertical change by horizontal change.

Sources & further reading

Reference this concept