7.1 Systems of Equations
A system of equations is a set of equations with a common solution.
Example:
S
| x2 - 3 x + 2 = 0
| x2 - x - 2 = 0
The first equation has solution x = 2 or x = 1,while the
second has solution x = 2 or x = -1. The common
solution is x = 2. Thus, that is the solution of the system.
Example:
S1
| x + xy = 1
| y - xy = - 4
This system equations has two unknown quantities. The solution will
consist of ordered pairs of numbers ( x, y ) satisfying
both equations.
A useful method for solving systems of equations is the method of
substitution. In this method, one solves one of the equations in the
system for a selected variable in terms of the other variables. Then all
occurances of the selected variable in the other equations of the system
are replaced by the solution obtained.
In the example at hand, the first equation (denoted E1 ) is solved for x in terms of y
to get
x = 1 / ( 1 + y )
The x in E2 is replaced with 1 / ( 1
+ y ) to produce an equation in only one variable, y. This
yields a second system of equations which is equivalent to the
first sytem of equations. Two systems of equations are equivalent if they
have the same solution.
S2
| x = 1 / ( 1 + y )
| y - y / ( 1 + y ) = - 4
The new E1 is equivalent to the original
E1 and the new E2 is equivalent to the original E2 given the original E1. Thus the new system of equations is
equivalent to the original system of equations.
Now, E2 can be replaced by the equivalent
equation ( y + 2 )2 / ( y + 1
) = 0, which, in turn is equivalent to the equation
y = - 2. Thus,
S3
| x = 1 / ( 1 + y )
| y = -2
is equivalent to the original system of equations. Using the principle of
substitution, E1 is equivalent to x
= -1. Thus the system of equations
S4
| x = - 1
| y = - 2
Is equivalent to the original system. That means it has the same solution
as the original system. But the solution of this last system is
transparent. Thus, we have solved the original system.
In this example we see a general approach to solving all systems of
equations. It consists of replacing one system by another equivalent
system in a systematic way in order to produce (eventually) an equivalent
system whose solution is obvious.
We have seen a couple of principles involved in replacing one system with
an equivalent system:
(1) Any equation in the system may be replaced by an equivalent equation.
(The Equivalent Equation Principle)
(2) Any variable or expression in an equation of the system may be
replaced by another expression found to be equivalent to that variable or
expression in one of the other equations of the system. (The Substitution
Principle)
There are other principles which allow us to find equivalent systems, some
of which will be discussed in the next section.
Exercise 7.1.1
Use the equivalent equation principle and the substitution principle to
solve the system of equations:
S1
| x + y = 0
| x + y2 = 0
Solution
Exercise 7.1.2
Solve the system of equations:
S1
| x + y = 1
| 2x + y = -1
Solution
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