Math 1030-003
Linear Modeling Worksheet
Name_________________________
In Math 1010, we learned about the point-slope equation for a line. Given two points,
(x1, y1), and (x2, y2), the slope between the points is given by
m=
y1 ! y2
x1 ! x2
The equation for the line that passes through these two points is given by
y ! y1 = m(x ! x1 )
Example 1:
Find the equation for the line between the points (2, 4), and ( !3, 1).
Solution:
First we need to find the slope between these two points. Using our formula with
(x1, y1) = (2, 4), and (x2, y2) = ( !3, 1), the slope is
m=
The formula for the line is
y1 ! y2
4 !1
3
=
=
x1 ! x2 2 ! (!3) 5
y ! y1 = m(x ! x1 )
y!4 =
3
( x ! 2)
5
Solving for y, we can write this equation as
y=
3
14
x+
5
5
If we plug in a value for x, we will find a corresponding value for y, so we say that y
depends on x. We call y the dependent variable, and we call x the independent
variable.
In real-life situations, we often encounter quantities that change over time. We can create
linear models of how these quantities are changing by viewing the changing quantity as
the dependent variable y, and time as the independent variable x.
Example 2:
Imagine that we are measuring rain fall with a rain gauge. After 1 hour, there is 1 inch of
rain in the gauge. After 5 hours, there are 6 inches of rain in the gauge. Create a linear
model to represent this situation.
Solution:
If we let x represent time (in hours), and we let y represent the amount of rain in the rain
gauge (in inches), then we have the following 2 data points:
(x1, y1) = (1, 1), and (x2, y2) = (5, 6)
The slope between these points is given by
m=
y1 ! y2 1 ! 6 !5 5
=
=
=
x1 ! x2 1 ! 5 !4 4
The linear model to represent this situation is the formula
y ! y1 = m(x ! x1 )
y !1=
5
(x ! 1)
4
Solving for y,
y=
5
1
x!
4
4
1.) Find the equation for the line passing through the points (1, 5), and (3, 10).
2.) You were 20 inches tall at birth, and 48 inches tall on your 8th birthday. Create a linear
equation to represent this situation. According to your equation, how tall will you be at
age 20? (Hint: plug in 20 for x, and solve for y)
3.) The population of Spring City was 5,000 in 1990, and 7,250 in 2000. Create a linear
equation to represent this situation. According to your equation, what will the population
be in the year 2006?
4.) The following table represents measurement of the height of a bean sprout:
Days after sprout was planted
Height of sprout in inches
5
1
7
9
2.5 4
11
5.5
Create a linear equation to represent this situation. According to your equation, how tall
will the sprout be after 14 days?