Unknown

Equations

4.1 Simultaneous Equations in Two or Three

Unknowns

A man standing on a railway bridge 32 meters long observes a

train coming towards the bridge at 120 km per hour. It turns

out that whichever way he runs at his top speed of 15 km per

hour, he will reach the end of the bridge at the same time as the

train. How far from the end of the bridge closest to the train is

he?

This typical textbook problem can be solved by the introduction of vari-
ables and setting up equations which relate them. For example, if we let x
be the distance to the end of the bridge nearest the train and y the dis-
tance to the other end in meters, we can derive the simultaneous system of
equations
x+y=32 y-x=4

both of which must be satisfied by x and y.
This system can be solved in a straightforward way. Using one equation,
we can solve for y in terms of x and use this to obtain from the other equa-
tion a single equation in x. However, some problems involve equations of
higher degree and it is not so easy to untangle the variables. In this section,
a few techniques for handling a simultaneous system will be reviewed. Un-
less otherwise specified, x, y, z will denote variables and the other letters
constants.

Exercises

1. Solve the system of equations

x+y=16

x+%=20

y + f = 22.

2. Consider the system

alx+bly+clz=O