2. Chapter 7: Systems of Equations and
Inequalities; Matrices
7.1 Systems of Equations
7.2 Solution of Linear Systems in Three Variables
7.3 Solution of Linear Systems by Row Transformations
7.4 Matrix Properties and Operations
7.5 Determinants and Cramer’s Rule
7.6 Solution of Linear Systems by Matrix Inverses
7.7 Systems of Inequalities and Linear Programming
7.8 Partial Fractions
Copyright © 2007 Pearson Education, Inc. Slide 7-2

3. 7.1 Systems of Equations
• A set of equations is called a system of equations.
• The solutions must satisfy each equation in the
system.
• A linear equation in n unknowns has the form
where the variables are of
a x a x a x b n n     1 1 2 2
first-degree.
• If all equations in a system are linear, the system is a
system of linear equations, or a linear system.
Copyright © 2007 Pearson Education, Inc. Slide 7-3

4. 7.1 Linear System in Two Variables
• Three possible solutions to a linear system in two
variables:
1. One solution: coordinates of a point,
2. No solutions: inconsistent case,
3. Infinitely many solutions: dependent case.
Copyright © 2007 Pearson Education, Inc. Slide 7-4

5. 7.1 Substitution Method
Example Solve the system.
Solution
x y (1)
3  2 
11
 x  y

3
(2)
y  x 
Solve (2) for y.
x x
3  2(  3) 
11
x x
3  2  6 
11
x
5 5
1
3


 
1 3
4

x
y
y
Substitute y = x + 3 in (1).
Solve for x.
Substitute x = 1 in y = x + 3.
Solution set: {(1, 4)}
Copyright © 2007 Pearson Education, Inc. Slide 7-5

6. 7.1 Solving a Linear System in Two
Variables Graphically
Example Solve the system graphically.
x y (1)
3  2 
11
 x  y

3
Solution Solve (1) and (2) for y.
(2)
Copyright © 2007 Pearson Education, Inc. Slide 7-6

7. 7.1 Elimination Method
Example Solve the system.
x y (1)
3  4 
1
x y
2  3 
12
(2)
Solution To eliminate x, multiply (1) by –2 and (2)
by 3 and add the resulting equations.
x y (3)
   
6 8 2
x y
6  9 
36
(4)
y
17 34
2

y

Copyright © 2007 Pearson Education, Inc. Slide 7-7

8. 7.1 Elimination Method
Substitute 2 for y in (1) or (2).
x
3  4(2) 
1
x
3 9
The solution set is {(3, 2)}.
3


x
• Check the solution set by substituting 3 in for x
and 2 in for y in both of the original equations.
Copyright © 2007 Pearson Education, Inc. Slide 7-8

9. 7.1 Solving an Inconsistent System
Example Solve the system.
x y (1)
3  2 
4
 6 x  4 y

7
(2)
Solution Eliminate x by multiplying (1) by 2 and
adding the result to (2).
Solution set is .
x y
6  4 
8
 x  y

6 4 7
0 15 Inconsistent System
Copyright © 2007 Pearson Education, Inc. Slide 7-9

10. 7.1 Solving a System with Dependent
Equations
Example Solve the system.
x y (1)
  
4 2
x y
8  2  
4
(2)
Solution Eliminate x by multiplying (1) by 2 and adding the
result to (2).
x y
  
8 2 4
x y
8  2  
4
0  0
Each equation is a solution of the other. Choose either equation
and solve for x.

4
x y x
 4   2
 
y
2
  
  

y
 
The solution set is e.g. y = –2:
2
y
, .
4


22    
{( , 2)} ( 1, 2)} 4
Copyright © 2007 Pearson Education, Inc. Slide 7-10

11. 7.1 Solving a Nonlinear System of
Equations
Example Solve the system.
x y (1)
3 2
 2 
5 x y
  
3 4
(2)
Solution Choose the simpler equation, (2), and
solve for y since x is squared in (1).
x y
  
3 4
 4


3
x
y
4x
Substitute for y into (1) .
(3)
3
Copyright © 2007 Pearson Education, Inc. Slide 7-11

12. 7.1 Solving a Nonlinear System of
Equations
5
  4

3
2
x
3 
2
2

 



x
x x
9  2(  4  ) 
15
2
x x
9  2  7 
0
x x x x
(9 7)( 1) 0 1 or
      
Substitute these values for x into (3).
 7
  
 
4 9
y  y
3
The solution set is
7
9
 
1
  
4 1
3
  or

43
27
 7 ,  43
,   1, 
1. 9
27
Copyright © 2007 Pearson Education, Inc. Slide 7-12

13. 7.1 Solving a Nonlinear System
Graphically
Example Solve the system.
(1)
(2)
y 
2
x
x y
  
2 0
Solution (1) yields Y1 = 2x; (2) yields Y2 = |x + 2|.
The solution set is {(2, 4), (–2.22, .22), (–1.69, .31)}.
Copyright © 2007 Pearson Education, Inc. Slide 7-13

14. 7.1 Applications of Systems
• To solve problems using a system
1. Determine the unknown quantities
2. Let different variables represent those
quantities
3. Write a system of equations – one for each
variable
Example In a recent year, the national average spent on two
varsity athletes, one female and one male, was $6050 for
Division I-A schools. However, average expenditures for a
male athlete exceeded those for a female athlete by $3900.
Determine how much was spent per varsity athlete for
each gender.
Copyright © 2007 Pearson Education, Inc. Slide 7-14

15. 7.1 Applications of Systems
Solution Let x = average expenditures per male
y = average expenditures per female
x y
 x y
   
6050 12,100

2
Average spent on
one male and one
female
(1)
(2)
12100
3900
x y
 
x y
 
2 x  16000
x  8000
Average Expenditure per male: $8000, and
per female: from (2) y = 8000 – 3900 = $4100.
Copyright © 2007 Pearson Education, Inc. Slide 7-15