Sec 3.1 Systems of Linear Equations in Two Variables
Learning Objectives:
1.
Deciding whether an ordered pair is a solution.
2. Solve a system of linear equations using the graphing, substitution, and elimination method.
1.
Deciding Whether an Ordered Pair Is a Solution
System of Equation
—consists of at least two or more linear equations.
Example.
=
−
−
=
−
0
2
12
3
4
.
1
y
x
y
x
=
+
=
−
+
=
+
−
0
2
2
2
5
.
2
z
x
z
x
y
z
y
x
Solution of the system
—is the point(s) where the graphs intersect (give true for both equations)
Example 1
. Determine whether the ordered pairs are a solution to the system:
=
+
=
−
6
2
3
4
2
y
x
y
x
a.
(
)
0
,
2
Answer:___________________
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b.
(
)
3
,
4
−
Answer:___________________
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2. Solve a system of linear equations using the graphing method
Three types of the System of Equations
.
1.
Consistent system with Independent equation
•
Two lines intersect at one point
(
)
y
x
,
.
•
Has one solution,
(
)
y
x
,
.
•
2
1
m
m
≠
•
When solve the system, get
x
= a number, y = a number.
2.
Inconsistent System
•
Two lines are parallel.
•
Has no solution,
φ
or
{ }
.
•
2
1
m
m
=
and
2
1
b
b
≠
•
When solve, get false statement.
3.
Consistent system with dependent equation
•
Two lines lie on top of the others (same line).
•
Has infinitely many solutions,
(
)
{
}
b
mx
y
y
x
+
=
,
or
(
)
{
}
c
by
ax
y
x
=
+
,
•
2
1
m
m
=
and
2
1
b
b
=
•
When solve the system, get true statement.
x
y
x
y
x
y

Steps to solve linear equations by graphing
1.
Solve and graph each equation separately.
2.
Identify type of systems (consistent, inconsistent, or dependent).
3.
State number of solution (one solution, infinitely many solutions or no solution).
Example 2.
Solve by graphing.
Label at least two points for each graph on the graph grid.
=
−
=
+
0
4
2
4
2
y
x
y
x
Solution:___________________
----------------------------------------------------------------------------------------------------------------------------------
3.
Solve a system of linear equations using the elimination method
Steps:
1. Write each equation in the form:
C
By
Ax
=
2.
Choose variable to eliminate.
3.
If necessary, multiply one or both equations by appropriate number(s) so that the coefficients of the
eliminated variable will have the sum of zero.
4.
Add two equations together.
5.
Solve for the variable.
6.
Solve for the other variable.
7.
State the final solution in ordered pair, if it exists.
Example 3.
Solve linear equations using the elimination method.
+
−
=
=
−
−
y
x
y
x
6
4
8
0
4
3
2
Answer:___________________________________
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