Download Chapter 2 Systems of Linear Equations and Matrices

Chapter 2
Systems of Linear Equations
and Matrices
Section 2.1
Solutions of Linear Systems
Solutions of First - Degree Equations

A solution of a first –degree equation
in two unknowns is an ordered pair,
and the graph of the equation is a
straight line.
Possible Solutions

Unique solution

Inconsistent System

Dependent System
Unique Solution



When the graphs of two first-degree
equations intersect, then we say the
point of intersection is the solution of
the system.
This solution is unique in that it is the
only point that the systems have in
common.
The solution is given by the coordinates
of the point of intersection.
Inconsistent Systems


When the graphs of two first-degree
equations never intersect (in other
words, they are parallel), there is no
point of intersection.
Since there is not a point that is
shared by the equations, then we
say there is no solution.
Dependent System


When the graphs of two first-degree
equations yield the exact same line,
we say that the equations are
dependent because any solution of
one equation is also a solution of the
other.
Dependent systems have an infinite
number of solutions.
Solving Systems of Equations

There are many methods by which a
system of equation can be solved:
• Graphing
• Echelon Method (using transformations)
• Substitution Method
• Elimination Method
Elimination Method


Try to eliminate one of the variables
by creating coefficients that are
opposites.
One or both equations may be
multiplied by some value in order to
get opposite coefficients.
Example 1

Solve the system below and discuss
the type of system and solution.
x + 2y = 12
-3x – 2y = -18
Example 1
x + 2y = 12
-3x – 2y = -18
-2x
= -6
x=3
Solve for y:
x + 2y = 12
3 + 2y = 12
2y = 9
y = 4.5
Example 1


Check x = 3 and y = 4.5 in other
equation.
-3x – 2y = -18
-3(3) – 2(4.5) = -18
-9 – 9 = -18
-18 = -18 √
Solution:
Unique solution: (3, 4.5)
Independent system
Example 2

Solve the system below and discuss
the type of system and solution.
2x – y = 3
6x – 3y = 9
Example 3

Solve the system below and discuss
the type of system and solution.
x + 3y = 4
-2x – 6y = 3
Example 4

Solve the system below and discuss
the type of system and solution.
4x + 3y = 1
3x + 2y = 2
Example 5

Solve the system below and discuss the
type of system and solution.
x + y= 6
5
5
x +y=5
10
3
6