#
Section 5-5

#
Problem Solving Using Linear Systems

Linear systems are used to model all kinds of real-world situations. (I have a geologist friend who uses computers to solve linear systems with hundreds of thousands of equations in hundreds of thousands of variables - all to try to figure out where earthquake-prone fault lines are.)

Here is a much, much simpler real-world application of linear systems. It sometimes takes some thought to translate the problem from English language to mathematical symbols, but once you're done with that, it's straightforward.

Example:

You drive 150 miles to a national park. It takes you three hours. Part of the time you’re on a freeway, where the speed limit is 70 mph. The rest of the time you’re on smaller country roads, where the speed limit is 30 mph. Supposing you drove exactly at the speed limit the whole way, how much time did you spend on each type of road?

The two unknowns,
*
x
*
and
*
y
*
, will stand for the amount of time spent on the freeway and the amount of time spent on the country roads.

Since the whole trip takes three hours, we have the first equation:

*
***
x
**
**
+
***
y
*
= 3

Now, since rate × time = distance travelled, 70
*
x
*
is the distance travelled on the freeway and 30
*
y
*
is the distance travelled on the country roads. Adding these, we have:

**
70
***
x
*
+ 30
*
y
*
= 150

We will solve the system by the
substitution method
. First, solve the first equation for
*
y
*
:

*
***
y
**
**
= 3 –
***
x
*

Now substitute 3 -
*
x
*
for
*
y
*
in the second equation:

**
70
***
x
*
+ 30(3 –
*
x
*
) = 150

Simplify and solve for
*
x
*
.

**
70
***
x
*
+ 90 – 30
*
x
*
= 150

**
40
***
x
*
= 60

*
***
x
**
**
= 1.5
**

Substituting back in
*
x
*
+
*
y
*
= 3, we have

*
***
y
**
**
= 1.5
**

So you spend one and a half hours driving on each type of road.