Review
Inconsistent &
Dependent Systems
Section 8.2
Definitions of Systems
Represented by Previous Examples
• A system of equations that has at least
one solution is said to be consistent.
• A system of different equations will yield
intersecting graphs (lines, planes, etc.)
and are said to be independent.
More Definitions of Systems
• A system that does not have a solution is
an inconsistent system.
• If at least one of the equations can be
expressed as a linear combination (or
equivalent form) of some of the other
equations in the system, then the system
is a dependent system. Graphically this
means that there is an overlapping of the
graphs in some way.
Example 1
The nutritional content per ounce of three foods is
presented in the table on the right. If a meal
consisting of the three foods allows exactly 2200
calories, 110 grams of protein, and 900 milligrams
of vitamin C, how many ounces of each kind of
food should be used?
Calories Protein
Vitamin C
(in grams) (in milligrams)
Food A
100
9
50
Food B
400
8
250
Food C
300
15
100
So,…Could you have systems with
more than one solution?
• Yes! In a situation where you have two
equations with three variables, then graphically
this is the intersection of two planes. If the
planes intersect to form a line, rather than a
point, there would be infinitely many solutions.
All points lying on the line would be solutions.
• You can’t state infinitely many points, so you
state the general form of all points on the
line, in terms of one of the variables (usually
the last one).
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Recognizing Dependent Systems from Matrices
• If the bottom row of your rref matrix has all 0’s, then the
system is dependent. This type of matrix also has an infinite
number of solutions, so it is a consistent system.
1 0 4 −6 
0 1 4 8 


0 0 0 0 
• Since it is impossible to write ALL the solutions, write the
general solution using one variable (Write both x and y in
terms of z). In other words, the x & y values are dependent
on the value selected for z.
• If there are infinitely many solutions, the system is
considered to be dependent.
Recognizing Inconsistent Systems from Matrices
• If the bottom row of your rref matrix has all 0’s and a 1,
then the system is inconsistent. The system is also
independent since all of the equations are non-equivalent.
• For example, in the matrix
1 0 4 −6 
0 1 4 8 


 0 0 0 9 
the last row corresponds to the false equation 0 = 9, so we
know the original system has no solution.
Example 2
Solve the system of equations stating the solution
appropriately. If there is no solution, so state.
Determine the types of systems represented by the
solution, as well.
x + y - 10z = - 4
x - 7z = - 5
3x + 5y - 36z = -10
Graphically, what is happening with
an inconsistent system?
• Recall, with 3 variables, the equation
represents a plane, therefore we are
considering the intersection of 3 planes.
• If a system is inconsistent, 2 or more of
the planes may be parallel. OR 2 planes
could intersect forming 1 line and a
different pair of planes intersect at a
different line, therefore there is nothing in
common to all three planes.
Example 3
Example 4
Solve the system of equations stating the solution
appropriately. If there is no solution, so state.
Determine the types of systems represented by the
solution, as well.
Solve the system of equations stating the solution
appropriately. If there is no solution, so state.
Determine the types of systems represented by the
solution, as well.
2x - 4y + z = 3
x - 3y + z = 5
3x - 7y + 2z = 12
x + 2y = 1
2x + 4y = 3
2
Example 5
Example 6
Solve the system of equations stating the solution
appropriately. If there is no solution, so state.
Determine the types of systems represented by the
solution, as well.
5x - 11y + 6z = 12
- x + 3y - 2z = - 4
3x - 5y + 2z = 4
Example 7
• Solve the system of equations stating the solution
appropriately. If there is no solution, so state.
Determine the types of systems represented by the
solution, as well.
2x + 3y – 3z = 7
5x + y – 4z = 2
4x + 2y - z = 6
Using The Graphing Calculator*
1. Press 2nd MATRIX to enter matrix mode.
2. Arrow over to EDIT, then choose the matrix you want to
use, then press ENTER.
3. Input the number of columns, press ENTER, then the
number of rows, press ENTER. The cursor should
move to the first entry in the matrix.
4. Input the entries row by row, from left to right, pressing
ENTER between entries.
5. When you finish, press 2nd QUIT to exit matrix mode.
6. Now, go back into matrix mode by pressing 2nd
MATRIX.
7. Arrow over to MATH, then down to “rref(“. Press
ENTER.
8. Press 2nd MATRIX, arrow down to the matrix you used,
then press ENTER.
9. The resulting matrix will be in reduced row echelon
form.
1 0 0 #
0 1 0 # 


0 0 1 #
*These directions are for a TI-83 and TI-84 only. Directions for TI-89
and Casio calculators are handed out in class.
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