Download Unit 7 PowerPoint Slides

EGR 1101 Unit 7
Systems of Linear Equations in
Engineering
(Chapter 7 of Rattan/Klingbeil text)
Systems of Linear Equations

A linear equation in one variable has a
unique solution.


A linear equation in two variables does
not have a unique solution.


Example: 2x=8 has a unique solution, namely x=4.
Example: 3x-4y=7 does not have a unique
solution.
But a system of two independent linear
equations in two variables does have a
unique solution.

Example: The pair of equations 3x-4y=7 and
2x+8y=26 has a unique solution, namely x=5 and
y=2.
Generalizing

More generally, for any positive
integer n, a system of n independent
linear equations in n variables does
have a unique solution.

It’s not unusual in engineering problems to end
up with, say, eight equations in eight variables.
Four Methods

We’ll study four methods for
attacking such problems:
1.
2.
3.
4.

Substitution
Graphical method
Matrix algebra
Cramer’s Rule (a shortcut derived from matrix
algebra)
For a given problem, all four
methods should give the same
solution!
Today’s Examples
1.
2.
Currents in a two-loop circuit
Forces in static equilibrium: Hanging
weight
A 2-by-2 Matrix Equation


Suppose we have the system of
equations
a11x1 + a12x2 = b1
a21x1 + a22x2 = b2
We can write this in matrix form as
 a11 a12   x1   b1 
 
a



 21 a22   x2  b2 
or
Ax=b
Rewriting a Matrix Equation



Suppose that in the matrix equation
Ax=b
A is a matrix of known constants,
and x is a vector of unknowns, and
b is a vector of known constants.
We can solve for the unknowns in x
by rewriting this equation as
x = A-1 b
The problem becomes: How do we
find the inverse matrix A-1?
Determinant of a 2-by-2 Matrix

Suppose we have a matrix A given by
 a11 a12 
A

a21 a22 


This matrix’s determinant is given by
|A|  a11a22  a12a21
We sometimes use the symbol  for the
determinant.
Inverse of a 2-by-2 Matrix

Suppose again we have a matrix A given
by
 a11 a12 
A

a21 a22 

This matrix’s inverse is given by
1
A 
A
1
 a22  a12 
 a

 21 a11 
Method 4. Cramer’s Rule

This shortcut rule says that the
solutions of a matrix equation A x = b
are given by:
x1 
A1
A
, x2 
A2
A
, ... , xn 
An
A
where Ai is obtained by replacing the ith
column of A with the vector b.
Solving Matrix Equations with
MATLAB


First, define our coefficient matrix and
our vector of constants:
>> A = [10 4; 4 12]
>> b = [6; 9]
MATLAB offers at least three ways to
proceed from here:
>> x = inv(A)*b
>> x = A^-1 * b
>> x = A \ b