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1
MA 15200
Lesson 32, Section 5.1
A linear equation with two variables has infinite solutions (many ordered pairs, example
6
3
2
=

y
x
)
However, if a second equation is also considered, and we want to know
what ordered pairs they may have in common, how many solutions may there be?
When
two linear equations are considered together, it is known as a
system of linear equations
or a linear system.
The
solution(s)
of the system is/are any
ordered pair(s)
they have
in common.
Consider graphing two lines.
How many points may they have in common?
As demonstrated in the graphs above, there are 3 situations.
1.
The graphs may intersect at one point.
If the lines of the equations intersect, the
solution is one ordered pair.
2.
The graphs may intersect at an infinite number of points.
If the lines of the
equations form the same line, the solution is an infinite number of ordered pairs.
3.
The graphs may not intersect.
If the lines of the equations do not intersect, there
is no solution.
Ex 1:
Given the ordered pair, determine if it is a solution of the shown system.
2
5
16
)
( 2,4)
3
2
5
2
1
3
)
,1
10
5
1
5
x
y
a
x
y
x
y
b
x
y
+
=

+
= 
+
=
+
=
If an ordered pair is a
solution of the system of
equations, it is a solution
of both equations.
In
other words, substituted in
both equations yields a
true statement.
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