7.3
Multivariate
Linear Systems
and Row
Operations
Copyright © 2011 Pearson, Inc.
What you’ll learn about
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Triangular Form 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.
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 3
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.
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 4
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.
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 5
Example Finding a Row Echelon Form
Solve the system:
2x + 3y - z = -1
-x + 5y + 3z = -10
3x - y - 6z = 5
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 6
Example Finding a Row Echelon Form
2x + 3y - z = -1
-x + 5y + 3z = -10
3x - y - 6z = 5
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 7
Example Finding a Row Echelon Form
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 8
Example Finding a Row Echelon Form
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
x + 10 - 3 = 10
(
so y = -2;
so x = 3.
)
The solution is 3, - 2,1 .
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 9
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.
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 10
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.
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 11
Example Solving a System Using
Inverse Matrices
Solve the system
2x - 3y = 0
2x - 2 y = 10
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 12
Example Solving a System Using
Inverse Matrices
Write the system as a matrix equation.
é 2 -3ù
éxù
é0ù
Let A = ê
ú , X = ê ú , and B = ê ú .
ë 2 -2 û
ë yû
ë10 û
é 2 -3ù é x ù é 2x - 3y ù
Then A× X = ê
ú×ê ú = ê
ú so that
ë 2 -2 û ë y û ë 2x - 2 y û
AX = B, where A is the coefficient matrix of the system.
A-1 exists since det A ¹ 0. Use grapher to find
é15 ù
X = A B = ê ú . The solution of the system is (15,10).
ë10 û
-1
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 13
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 = ax 2 + bx + c.
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 14
Example Fitting a Parabola to Three Points
(
) ( )
(
Determine a, b, and c so that -3, 32 , 1, 4 and 5, 40
()
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
)
are on the graph of f x = ax + bx + c.
2
é 9 -3 1ù
éaù
é 32 ù
ê
ú
ê ú
ê ú
A = ê 1 1 1ú , X = ê b ú , and B = ê 4 ú
êë 25 5 1úû
êë c úû
êë 40 úû
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 15
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 = ax + bx + c.
2
A grapher shows that
é2ù
ê ú
-1
X = A B = ê -3ú .
êë 5 úû
Thus a = 2, b = -3, and c = 5.
()
The graph of the quadratic function f x = 2x 2 - 3x + 5
contains the three points ( - 3, 32), (1, 4) and (5, 40).
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 16
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 = ax + bx + c.
2
Support Graphically
The figure shows a graph
of y1 = 2x 2 - 3x + 5
superimposed on a scatter
plot of the three points.
The points appear to lie
on the curve.
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 17
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) = x 3 - 4x - 1?
4. Solve for x in terms of the other variables:
x + z + w = 2
é 2 1ù
5. Find the inverse of the matrix ê
ú.
ë 0 3û
Copyright © 2011 Pearson, Inc.
Slide 7.3 - 18
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) = x - 4x - 1? yes
3
4. Solve for x in terms of the other variables:
x + z + w = 2
x = 2- z- w
é 2 1ù
5. Find the inverse of the matrix ê
ú
ë 0 3û
Copyright © 2011 Pearson, Inc.
é1/2 -1 / 6 ù
ê
ú.
1/ 3 û
ë0
Slide 7.3 - 19