# Math 308: Matrix Algebra Chapter 1.1 Notes

# 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:

- 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: $10300~~*0.95*0.05=489.25$

~~Federal tax: $10300~~*0.95*0.95*0.4=3718.3$

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:

- If another system of equations has same solution with one but have different parameters, it means geometrically they have
**same point of intersection**.

## Echelon form example:

Solution:

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

## 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**: