Algebra:
Simultaneous Equations
For example,
x  y  1

x  y  0
is a system of simultaneous equations. Solving such a system means finding,
if possible, the values of x and y for which the two equations are
simultaneously satisfied.
From the second line we see that x  y , so that we can rewrite the system as
follows:
y
x
,

x  y  1
which gives
x y

2 y  1
and finally
 x  1/ 2
.

 y  1/ 2
The system has a unique solution ( x, y )  (0.5, 0.5) .
If you want to practice on a more complicated system, try this one:
x  2 y  z  1

2 y  z  1 .

x
z0

 x  2/3

Solution:  y  1 / 6 .
 z  2/3

Sometimes a system of simultaneous equations can have more than one
solution. Consider the system
2 x  2 y  0
.

x y0
It is not hard to see that each pair of numbers ( x, y ) such that x  y satisfies
the system. The system has therefore infinitely many solutions.
Try to solve this one:
x y z0

x y z0 .
2 x
 2z  0

Solution: Each triplet ( x, y, z ) such that x   z and y  0 satisfies the system.
Let’s modify the system above a little:
 x  y  z 1

 x  y  z 1 .
2 x
 2z  0

This time the triplet ( x, y, z ) would have to satisfy x   z, y  1, y  1, which is
impossible.
The last system therefore has no solution.