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Unit 5. Systems of Equations and Inequalities
In Units 1 to 4 our study of algebra and trigonometry has been dealing with either a function of
one variable or a single equation (or inequality) in two variables.
However, many problems in many
different fields of study give rise to two or more equations (or inequalities) in two or more variables.
To
solve such problems
, we need to find all the
solutions
of a
system of equations
(or inequalities)
.
By a
solution to a system of equations (or inequalities), we mean the values of the variables that satisfy each
equation (or each inequality) in the system.
For this last unit of Math 17 we will focus on methods for finding solutions that are common to
all equations in a system.
Of particular importance are the techniques involving
matrices
, because they
are well-suited for computer implementation and can be readily applied to a system containing any
number of linear equations in any number of variables.
We shall also consider systems of inequalities and
linear programming
, a powerful mathematical tool of major importance in solving a variety of practical
problems.
After completing this unit, you should be able to:
1.
Solve systems of nonlinear equations in two unknowns;
2.
Solve systems of linear equations in two and three unknowns;
3.
Solve systems of inequalities,
4.
Solve linear programming problems,
5.
Use systems of equations and inequalities to model and solve real-life problems.
OUTLINE:
1.
Systems of Equations in Two Unknowns: Substitution; Elimination
2.
Systems of Linear Equations in Three Unknowns: Gaussian Elimination
3.
Matrices and Determinants for Solving Linear Systems
4.
Systems of Inequalities
5.
Linear Programming

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Unit 5. Systems of Equations and Inequalities
Section 1
page 2
_____________________________________________________________________________________
5.1
Systems of Equations in Two Unknowns
No doubt you learned how to solve a linear system in high school. Let us have some recall.
A
linear system of two equations in two variables
, say
x
and
y
, is of the form
2
2
2
1
1
1
c
y
b
x
a
c
y
b
x
a
where
a
1
,
a
2
,
b
1
,
b
2
, c
1
,
and
c
2
are real numbers.
Here the brace is used to indicate that
the equations are to be treated simultaneously.
This system can be represented graphically as a pair of lines that are intersecting,
coincident, or parallel.
Example 5.1.1
Examples of Systems of Equations
in Two Unknowns
Systems of linear equations
Systems of nonlinear equations
Note that in a nonlinear system, one of your equations can be linear, just not all of them.
Solving Systems of Equations in Two Unknowns
In general, a solution of a
system of two equations in two variables
is an ordered pair that makes
BOTH equations true. Geometrically, it is where the graphs of the two equations intersect, what
they have in common.
In this section we will revisit solving systems of two equations in two variables

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