Unit 7.3

7.3
Multivariate
Linear Systems
and Row
Operations
Copyright © 2011 Pearson, Inc.

What you’ll learn about
 Triangular Forms for Linear Systems
 Gaussian Elimination
 Elementary Row Operations and Row Echelon Form
 Reduced Row Echelon Form
 Solving Systems with Inverse Matrices
 Applications
… and why
Many applications in business and science are modeled
by systems of linear equations in three or more
variables.
Copyright © 2011 Pearson, Inc. Slide 7.3 - 2

Equivalent Systems of Linear Equations
The following operations produce an equivalent
system of linear equations.
1. Interchange any two equations of the system.
2. Multiply (or divide) one of the equations by
any nonzero real number.
3. Add a multiple of one equation to any other
equation in the system.

Row Echelon Form of a Matrix
A matrix is in row echelon form if the following
conditions are satisfied.
1. Rows consisting entirely of 0’s (if there are
any) occur at the bottom of the matrix.
2. The first entry in any row with nonzero
entries is 1.
3. The column subscript of the leading 1 entries
increases as the row subscript increases.

Elementary Row Operations on a Matrix
A combination of the following operations will
transform a matrix to row echelon form.
1. Interchange any two rows.
2. Multiply all elements of a row by a nonzero
real number.
3. Add a multiple of one row to any other row.

Example Finding a Row Echelon Form
Solve the system:
2x  3y  z  1
x  5y  3z  10
3x  y  6z  5

2x  3y  z  1
x  5y  3z  10
3x  y  6z  5
Apply elementary row operations to find a row echelon
form of the augmented matrix.

2 3 1 1


1 5 3 10

3 1 6 5






uuRuu2r1

1 5 3 10
2 3 1 1
3 1 6 5









2R1  Ruuuuuuuuur2

1 5 3 10
0 13 5 21
3 1 6 5









3R1  Ruuuuuuuuur3

1 5 3 10
0 13 5 21
0 14 3 25









1
13
R2 uuuuur

1 5 3 10
0 1
5
13
21
13
0 14 3 25












uu1u4uuRu2uuuRuur3
1 5 3 10
0 1
5
13
21
13
0 0
31
13
31
13
















13
31
R3 uuuuuur
1 5 3 10
0 1
5
13
21
13
0 0 1 1













Solve the system:
2x  3y  z  1
x  5y  3z  10
3x  y  6z  5
Convert the matrix to equations and solve by substitution.
z  1;
y  5 / 13  21/ 13 so y  2;
x 10  3  10 so x  3.
The solution is 3,  2,1.

Reduced Row Echelon Form
If we continue to apply elementary row
operations to a row echelon form of a matrix, we
can obtain a matrix in which every column that
has a leading 1 has 0’s elsewhere. This is the
reduced echelon form.

Invertible Square Linear System
Let A be the coefficient matrix of a system of n
linear equations in n variables given by AX = B,
where X is the n  1 matrix of variables and B is
the n  1 matrix of numbers on the right-hand
side of the equations. If A–1 exists, then the
system of equations has the unique solution
X = A–1B.

Example Solving a System Using
Inverse Matrices
Solve the system
2x  3y  0
2x  2y  10

Example Solving a System Using
Inverse Matrices
Write the system as a matrix equation.
Let A 
2 3
2 2

 

 
, X 
x
y

 

 
, and B 

0
10
 

 
.
Then A X 
2 3
2 2

 

 

x
y

 

 

2x  3y
2x  2y

 

 
so that
AX  B, where A is the coefficient matrix of the system.
A-1 exists since det A  0. Use grapher to find
X  A-1B 

15
10
 

 
. The solution of the system is (15,10).

Example Fitting a Parabola to Three Points
Determine a, b, and c so that 3, 32, 1, 4 and 5, 40
are on the graph of f x ax2  bx  c.

We must have f 3 32, f 1 4, and f 5 40
f 3 9a  3b  c  32
f 1 a  b  c  4
f 5 25a  5b  c  40
A 

9 3 1
1 1 1
25 5 1









, X 
a
b
c










, and B 
32
4
40











A grapher shows that
X  A1B 

2
3
5









.
Thus a  2, b  3, and c  5.
The graph of the quadratic fucntion f x ax2  bx  c
contains the three points (Ğ3, 32), (1, 4) and (5, 40).

Support Graphically
The figure shows a graph
of y1  2x2  3x  5
superimposed on a scatter
plot of the three points.
The points appear to lie
on the curve.

Quick Review
1. Find the amount of pure acid in 45L of a 58%
acid solution.
2. Find the amount of water in 30 L of a 28%
acid solution.
3. Is the point (0, 1) on the graph of the function
f (x)  x3  4x 1?
4. Solve for x in terms of the other variables:
x  z  w  2
5. Find the inverse of the matrix

2 1
0 3

 
 
.

Quick Review Solutions
1. Find the amount of pure acid in 45L of a 58%
acid solution. 26.1 L
2. Find the amount of water in 30 L of a 28%
acid solution. 21.6 L
3. Is the point (0, 1) on the graph of the function
f (x)  x3  4x 1? yes
4. Solve for x in terms of the other variables:
x  z  w  2 x  2  z  w
5. Find the inverse of the matrix

2 1
0 3

 
 

1/2 1/ 6
0 1/ 3
 

 
.

Unit 7.3

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Unit 7.3