# Section 2.1 Lecture Notes MATH 141-501 Linear Systems: ```Section 2.1 Lecture Notes
MATH 141-501
Linear Systems:
There are many times where we need to solve systems
of linear equations:
Example: Solve
2x + 3y = 7
3x − 15y = 1.5
If the system has two variables, this is really the same
as:
1
There are three possibilities.
Only one solution:
2
No solution:
3
Infinitely many solutions:
4
Example: One Solution
Solve the system
x + 51y = 60
−x + y = 980
5
Example: Infinitely Many Solutions
Solve the system
5x + 10y = 1
1
1
y =− x+
2
10
6
No solution:
3x − y = 10
9x − 3y = 115
7
More Than Two Variables
Can we solve a system like
2x + 3y − 7z = 1
x + y + z = 121
4x − 5.5y + 13z = 3.4?
8
Geometric Interpretation of A System with Three
Variables
In the case of three variables, we are dealing with
The equation of a
Cz = D.
is Ax+By+
9
Systems of Three Variables
How many solutions can a system of linear equations
with three variables have?
Figure 1: Image from Finite Mathematics for the Managerial, Life, and Social
Sciences, Tan, 11e
10
Who cares? We do!
In terms of real-world applications of math, solving
systems of linear equations is one of the biggest.
The systems are generally
It is not uncommon for businesses to need to solve
systems involving
variables
11
Example:
A music hall has 1800 seats. They offer three ticket
prices: child(\$5), adult(\$15), and senior(\$12.50). At a
certain concert, the number of children and seniors combined was twice the number of adults. The performance
was sold out, and revenue from ticket sales was \$23,250.
How many tickets of each type did the music hall sell?
12
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