Class 1

Chapter 1.1

Ex.1 A discount movies sells 100 tickets Child tickets cost 3 dollars and adult tickets cost 5 dollars. If the total revenue is $422, how many tickets of each type were sold?

x = number of child tickets sold
y = number of adult tickets sold
x + y = 100
3x + 5y = 422

-> combine two equations 
-> solve
-> check x = 39 and y = 61 is a solution by substituting into both equations

Geometrical meaning:


Ex. 3 Find all solutions to the system of equations

3x + 6y = 10
6x + 12y = 15

Two equations are parellal to each other

Ex. 2 Find all solutions to the system of equations

4x + 10 y = 14
-6x - 15y = -21
The relationship between x and y is the same for both equations

To avoid confusion, we use another variable s (free variable)
-> x = -5s/2 + 7/2
-> y = s
Any choice of y corresponding value of x gives a solution
    
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If we selected for y instead:
y = -2x/5 + 7/5
-> x = t
-> y = -2t/5 + 7/5
-> (x, y) = (t, -2t/5 + 7/5) for any choice of t
Infinitely many solutions

Note: they are in the same line


System of 3 (more) variables The gemetry in R^3 is more complocated

3x + 2y + z = 6




Class 2


Conceptual Problems Chapter 1

(Wrong approach):

Bonuses: $10300*0.05=515$

State tax: $103000.950.05=489.25$

Federal tax: $103000.950.95*0.4=3718.3$

Solution:

$ x = $bonuses
$ y = $ state tax
$ z = $ federal tax

$ x = 0.05(103000-y-z)\\
y = 0.05(103000-x)\\
z = 0.40(103000-x-y)$

Result = $ \begin{cases} x = 3000\\
y = 5000\\
z = 38000 \end{cases} $


Triangular system example:

Echelon form example:

Solution:
$ (2+s, -1+s, s, 3) $

1.1 Textbook Notes