19-08-2013, 04:51 PM
ASSIGNMENT ON System of Equations
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Introduction and Summary:-
Until now, we have been
working with single equations. We have seen equations with one variable, which
generally have a finite number of solutions, and equations with two variables,
which are usually satisfied by an infinite number of ordered pairs. In this section,
we will begin to deal with systems of equations; that is, with a set of two or more
equations with the same variables. We limit our discussion to systems of linear
equations, since our techniques for solving even a single equation of higher
degree are quite limited.
Systems of linear equations can have zero, one, or an infinite number of
solutions, depending on whether they are consistent or inconsistent, and whether
they are dependent or independent. The first section will explain these
classifications and show how to solve systems of linear equations by graphing.
The second section will introduce a second method for solving systems of linear
equations--substitution. Substitution is useful when one variable in an equation
of the system has a coefficient of 1 or a coefficient that easily divides the
equation.
If one of the variables has a coefficient of 1 , substitution is very useful and easy
to do. However, many systems of linear equations are not quite so neat, and
substitution can be difficult. The third section introduces another method for
solving systems of linear equations--the Addition/Subtraction method.
BUSINESS MATHEMATICS & STATISTICS
Systems of Equations:
We have worked with two types of
equations--equations with one variable and equations with two variables. In
general, we could find a limited number of solutions to a single equation with one
variable, while we could find an infinite number of solutions to a single equation
with two variables. This is because a single equation with two variables is
underdetermined--there are more variables than equations. But what if we added
another equation?
A system of equations is a set of two or more equations with the same
variables. A solution to a system of equations is a set of values for the variable
that satisfy all the equations simultaneously. In order to solve a system of
equations, one must find all the sets of values of the variables that constitutes
solutions of the system.
Solving Systems of Linear Equations by Graphing:-
When we graph a
linear equation in two variables as a line in the plane, all the points on this line
correspond to ordered pairs that satisfy the equation. Thus, when we graph two
equations, all the points of intersection--the points which lie on both lines--are
the points which satisfy both equations.
To solve a system of equations by graphing, graph all the equations in the
system. The point(s) at which all the lines intersect are the solutions to the
system.