30

––

2

45

––

3

slope 467

No. widgets sold 0 1 2 3 4 5

Profit (in dollars) 0 15 30 45 60 75

Slope

g

The relationship is certainly linear, as each addi-

tional widget sold produces the same increase of $15 in

profit made. This relationship has slope 15. Notice that

in selling two additional widgets, profit is increased by

$30, and in selling three additional widgets, profit

increases by $45, and so on. In particular, the ratio of

profit increase to production increase is always the

same: = = 15. In general, if two quantities xand

yare linearly related, the slope of the relationship is

computed as:

In particular, if the value x1produces the output value

y1and x2the output y2, then:

2. Graphically: Plotting the points of a linear rela-

tionship produces a straight-line graph. The slope

of the line is determined by placing the tip of a

pencil at any location on the line, moving it hori-

zontally one unit to the right, and measuring the

vertical distance the pencil tip must move in order

to return to the line. An upward vertical motion is

considered positive and a downward vertical

motion negative. Graphing the points of the wid-

get/profit example again shows that this linear

relationship has slope 15.

It does not matter which point on the line one ini-

tially chooses: all locations yield the same value for the

slope via this method. (This is only true for straight-

line graphs.) One can also make a horizontal run of a

length different than 1 to produce a different value for

the rise one must take to return to the graph. The ratio

“rise over run,” however, will always adopt the same

value. School students are often asked to memorize this

catch phrase as the definition of slope:

If (x1, y1) and (x2, y2) are any two points on the line,

then we can interpret the quantity x2– x1as a run with

corresponding rise y2– y1and so, again, we see that

slope is given by:

A line has positive slope if, in moving from left to

right, the line rises in value, negative slope if it

decreases in value (thus a unit step horizontally to the

right requires a negative rise in order to return to the

line), and zero slope if it is horizontal.

Traffic highway signs often use a version of this

interpretation of slope to warn drivers of the steepness

of the road. For example, a sign that reads “15% grade

next 7 miles” warns drivers that for each mile traveled

for the next 7 miles, the altitude of the road drops by

0.15 miles. (The slope of the road is thus –0.15.)

3. Algebraically: The

EQUATION OF A LINE

is usually

written in the form y= mx + b, where mand bare

numbers. We see that an increase of the x-value by 1

unit causes an increase in the y-value by munits:

If x→x+ 1, then y= mx + b→m(x+ 1) + b

= (mx + b) + m= y+ m.

Thus the slope of the line is m. We have:

The slope of a line is simply the coefficient of

the x-variable in the equation y= mx + b.