6.1 Systems of Linear Equations in Two Variables Hwk: 15, 19, 21, 28, 43, 44
This lesson is a review from beginning and intermediate alg
We'll mention a couple points and assign minimal hwk
You review it as necessary
What does a system look like, algebraically
Solution techniques
Substitution (also work for nonlinear systems)
Elimination
Three cases of solutions (graphically)
Algebraically, what do these cases look like? (for the inconsistent and dependent cases)
New: Writing a solution for the dependent case Example 6
Cost/Revenue/Break-Even
Example: do #45 but build the revenue function)
6.2 Large Systems of Linear Equations Hwk: Std, -21, +20
Work toward triangular form, then finish by back-sub
X1-5, practice on back-sub
Elementary operations:
Swap any two equations
Mult an eqn by a const [not 0] -- replace the row by the result
Mult an eqn by a const and add to another eqn --
leave the 1st eqn unchanged, but replace the 2nd eqn
Strategy:
Pick a variable and eliminate the same var from all but one eqn.
Which eqn to keep? The one with the smallest coeff of the var (1 if possible)
You need the lower coeff to be a divisor of the higher ones; how:
divide the master eqn, or multiply the target eqn.
Result: system with one less var and one less eqn.
Repeat until you have triangular form.
6.3 Matrix Solution Methods Hwk: Std, +43, -41
Syst of eqns, converted to (and from) a matrix
Calculator entries: for most: 2nd, MATRIX, EDIT, pick a name A, enter dimension, enter matrix, quit and view the matrix on the home screen
Gaussian elim vs Gauss-Jordan
ref vs rref on calc
partial fraction example:
text already has problem set up
Example: X44
6.4A Matrix Algebra Hwk: Std to #41
Dimension
Double subscript notation: Meaning of aij, e.g. a23
Equality, +, -
Scalar mult
Properties of addition and scalar mult: same as for real numbers
[Do 6.4 before the matrix mult section here, except be able to do it on a calculator]
Properties of matrix mult: associative and distributive OK: same as real numbers, except:
product may not even exist (problem with closure)
not commutative
identity only applies to square matrices
inverse may not exist
6.4 Matrix Methods for Square Systems Hwk:
Return to: Merced College; Don Power Updated 08/16/05 by Don Power