1. Systems of Linear Equations in Engineering
EGR 1101 Unit 7
Systems of Linear Equations in Engineering
(Chapter 7 of Rattan/Klingbeil text)

2. Systems of Linear Equations
A linear equation in one variable has a unique solution.
Example: 2x=8 has a unique solution, namely x=4.
A linear equation in two variables does not have a unique solution.
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.

3. 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.

4. Four Methods We’ll study four methods for attacking such problems:
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!

5. Today’s Examples Currents in a two-loop circuit
Forces in static equilibrium: Hanging weight

6. 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 or
A x = b

7. Rewriting a Matrix Equation
Suppose that in the matrix equation
A x = 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?

8. Determinant of a 2-by-2 Matrix
Suppose we have a matrix A given by
This matrix’s determinant is given by
|A|  a11a22  a12a21
We sometimes use the symbol  for the determinant.

9. Inverse of a 2-by-2 Matrix
Suppose again we have a matrix A given by
This matrix’s inverse is given by

10. Method 4. Cramer’s Rule
This shortcut rule says that the solutions of a matrix equation A x = b are given by: where Ai is obtained by replacing the ith column of A with the vector b.

11. 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