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
z0
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 y0
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 z0
x y z0 .
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.
0
