The
Meaning of Slope and y-Intercept in
the Context of Word Problems

In the equation
of a straight line
(when the equation is written as "y
= mx + b"),
the slope
is the number "m"
that is multiplied on the x,
and "b"
is the y-intercept,
where the line crosses the y-axis.
This useful form of the line equation is sensibly named the "slope-intercept
form". Graphing
from this format
can be quite straightforward, particularly if the values of "m"
and "b"
are relatively simple numbers (such as 2
or –4.5,
rather than
^{17}/_{19}
or 1.67385).
In this lesson, we are going to look at the "real world" meanings
that slope and y-intercept
can have.

We have seen that
the slope of a line measures how much the value of y
changes for every so much that the value of x
changes. For instance, in the line y
= ( ^{3}/_{5} )x –
2, the slope is m
= ^{3}/_{5}.
This means that, starting at any point on this line, you can get to another
point on the line by going up 3
units and then going to the right 5
units. But we could also view this slope as a fraction over 1:

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In other words, for every
one unit that x
moves over to the right, y
goes up by three-fifths of a unit. While this doesn't necessarily graph
as easily as "three up and five over", it can be a more useful
way of viewing things when you're doing word problems.

Often, linear-equation
word problems deal with changes over the course of time; the equations
will deal with how much something changes as time passes. For instance,
an exercise might deal with how the population grows in a certain city,
with the population increasing by a certain fixed amount every year.

When x
= 0, the corresponding
y-value
is the y-intercept.
In the particular context of word problems, the y-intercept
(that is, the point when x
= 0) also refers to
the starting value; that is, the value when you started taking your reading
or started "counting" the time and its changes. In the example
from above, the y-intercept
would be the population when the sociologists started keeping track of
the population. If they started taking their measurements or doing their
calculations from a "base" year of 1980, then "x
= 0" would correspond
to "the year 1980", and the y-intercept
would correspond to "the population in 1980".

The following are a few
examples that further demonstrate how this works:

The average lifespan
of American women has been tracked, and the model for the data is y
= 0.2t + 73,
where t
= 0 corresponds to
1960. Explain the meaning of the slope and y-intercept.

What is the meaning of
the slope? It means that, every
year, the average lifespan of American women increased by
0.2 years, or about 2.4
months.

When
t = 0, what
is the value of y?
Looking at the equation, I see that y
= 73.

What is the meaning of
this y-value?
It means that, in
1960
(when they started counting), the average lifespan of an American woman
was 73
years.

The equation for
the speed (not the height) of a ball that is thrown straight up in the
air is given by v
= 128 – 32t,
where v
is the velocity (in feet per second) and t
is the number of seconds after the ball is thrown. With what initial
velocity was the ball thrown? What is the meaning of the slope?

What is the slope? It
is m
= –32. This value
tells me that, for every increase by 1
in my input variable
t,
I get a decrease of 32
in my output variable v.

What is the meaning of
the slope? It means that, every
second, the speed decreases by 32
feet per second. Eventually the velocity becomes zero (when the ball
reaches its peak), and then becomes negative (when gravity takes over
and pulls the ball back down to the ground).

When
t = 0, what is
the value of v?
Looking at the equation, I see that v
= 128. The exercise
defines v
as measuring the velocity of the ball.

What is the meaning of
this v-value?
It means that, when the ball was released at t
= 0 seconds (when
they started counting), it
was launched upward at 128
feet per second.

Fisherman in the
Finger Lakes Region have been recording the dead fish they encounter
while fishing in the region. The Department of Environmental Conservation
monitors the pollution index for the Finger Lakes Region. The model
for the number of fish deaths "y"
for a given pollution index "x"
is y
= 9.607x + 111.958.
What is the meaning of the slope? What is the meaning of the y-intercept?

What is the slope? It
is m
= 9.607. This value
tells me that, for every increase by 1
in my input variable x,
I get an increase of 9.607
in my output variable y.

What is the meaning of
the slope? In means that, for
every increase in the pollution index by one unit (say, from a pollution
index of 6
to a pollution index of
7), there are nine
or ten more fish deaths during the year.

When x
= 0, what is the
value of y?
Looking at the equation, I see that y
= 111.958.

What is the meaning of
this y-value?
It means that, even
if the index were zero (that is, even if the water were utterly pure),
there would still be about 112
fish deaths a year anyway.

Word problems with linear
(straight-line) equations almost always work this way: the slope is the
rate of change, and the y-intercept
is the starting value.

Stapel, Elizabeth.
"The Meaning of Slope and y-Intercept in the Context of Word
Problems." Purplemath. Available
from http://www.purplemath.com/modules/slopyint.htm.
Accessed