This document discusses systems of linear equations and methods for solving them. It defines a linear system as a set of equations where all variables have an exponent of 1. There are three possibilities for a system: 1) a single solution, 2) no solution (inconsistent), or 3) infinitely many solutions. Four methods are presented for solving systems: substitution, elimination, graphing, and matrices. Examples are provided to illustrate substitution and elimination. The document also discusses how to determine if a system is inconsistent or has infinitely many solutions based on the outcome of solving the system.

1. 9.1 Systems of Linear Equations
Chapter 9 Systems and Matrices

2. Concepts and Objectives
 Systems of Linear Equations
 Review solving systems by substitution and
elimination
 Determine whether a system is inconsistent or has
infinitely many solutions.

3. Systems of Linear Equations
 A set of equations is called a system of equations. If all of
the variables in all of the equations are of degree one,
then the system is a linear system. In a linear system,
there are three possibilities:
 There is a single solution that satifies all the
equations.
 There is no single solution that satisfies all the
equations.
 There are infinitely many solutions to the equations.

4. Systems of Linear Equations
 There are four different methods of solving a linear
system of equations:
1. Substitution – Solve one equation for one variable,
and substitute it into the other equation(s).
2. Elimination – Transform the equations such that if
you add them together, one of the variables is
eliminated. Then solve by substitution.
3. Graphing – Graph the equations, and the solution is
their intersection.
4. Matrices – Convert the system into one or two
matrices and solve.

5. Substitution
Example: Solve the system.
1. Solving for y in the second equation will be simplest.
2. Now replace y with x + 3 in the first equation and solve
for x.
3 2 11
3
x y
x y
 
  
3y x 
 3 2 11
3 2 6 11
5 5
1
3xx
x x
x
x
 
  




6. Substitution (cont.)
3. Now we replace x in the first equation with 1 and solve
for y.
4. The solution is the ordered pair 1, 4. It is usually a
good idea to check your answer against the original
equations.
 
4
1 3y
y
  

   
 
3 1 2 4 11
1 4 3
 
  

7. Elimination
Example: Solve the system.
While either variable would be simple to eliminate,
multiplying the second equation by 4 would be simplest.
4 3 13
5
x y
x y
  
  
4 3 13
4 4 20
x y
x y
  
  
7 7
1
y
y


Be sure to multiply
both sides!
1 5
4
4
x
x
x
  
 
   4,1

8. Inconsistent Systems
 If after solving for the variables you end up with a false
statement (for example, 0 = 9), this is an inconsistent
system in that it has no solutions.
Example: Solve the system.
3 2 4
6 4 7
x y
x y
 
  
6 4 8
6 4 7
0 15
x y
x y
 
  


9. Infinitely Many Solutions
 If after solving the variables you end up with a true
statement (ex. 0 = 0), then you have infinitely many
solutions. In this case, you would express your solution
set in terms of one of the variables, usually y.
Example: Solve the system.
4 2
4 2
0 0
x y
x y
  
  

4 2
2
4
x y
y
x
 


2
,
4
y
y
 
 
 

10. Classwork
 College Algebra & Trigonometry
 Page 848: 8-48 (4); page 788: 24-36 (even);
page 776: 6-10