# Class 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: • Both equations represent lines in R^2 • They intercept at a single point which is the solution 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 ======= 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\\

## 1.1 Textbook Notes • If a linear system has at least one solution, then we say that it is consistent. If not, then it is inconsistent. • a variable that appears as the first term in at least one equation is called a leading variable:  $※※※※※※※※※※※️$ $※※※※※※※※※※※️$