1. MATH 1046 Introduction to Systems of Linear Equations (Sections 2
MATH 1046 Introduction to Systems of Linear Equations (Sections 2.1 and 2.2)
Alex Karassev

2. Example: Error in scale
Alex is using a scale that is known to have a constant error. A can of soup and a can of tuna are placed on this scale, and it reads 720 grams. Four identical cans of soup and three identical cans of tuna are placed on an accurate scale, and a weight of 2.4 kg is recorded. If two cans of tuna weigh 540 grams on this bad scale, what is the amount of error in the scale and what is the correct weight of each type of can?

3. Mathematical Model
4 cans of soup and 3 cans of tuna on good scale equals 2.4 kg
A can of soup and a can of tuna on bad scale equals 720 g
2 cans of tuna on bad scale equals 540 g
Let x, y, and z be the weights of a can of soup, a can of tuna, and the error of the scale in grams, respectively. Then we obtain the following system of linear equations:

4. Example: Logging
A logging company has a contract to provide 1000 m³ of pine, 800 m³ of spruce, and 600 m³ of fir logs per month. There are three regions available for logging. The following table gives the species mix, and timber density for each region.
How many hectares should one log in each operating region listed above to deliver exactly the required volume of logs?
Region
Volume/hectare
% Pine
% Spruce
% Fir
West
330 m³
70 %
20 %
10 %
North
390 m³
60 %
30 %
East
290 m³
5 %
75 %

5. Mathematical Model
Region
Volume/hectare
% Pine
% Spruce
% Fir
West
330 m³
70 %
20 %
10 %
North
390 m³
60 %
30 %
East
290 m³
5 %
75 %
We need: 1000 m³ pine, 800 m³ spruce, and 600 m³ fir
Let w, n, e be the number of hectares logged in West, North, and East regions respectively. Then:

6. Modern applications of linear systems
Linear systems with hundreds (or more!) unknowns:
Applications in economics (linear optimization (programming), transportation problem, linear models)
Computer graphics
Optimization
Traffic flow

7. How to solve linear systems?
Straightforward approach: express one of the variables form one of the equations and substitute in the rest of equations
Continue until you get to one equation with one variable

8. Error in scale: solution
From last equation: z = 540 – 2y
Substitute in the first, we get x+ y + (540 – 2y) = 720
So, we get two equations: x – y = 180 4x + 3y = 2400
Now from the first equation x = y and therefore 4(180 + y) + 3y = 2400
Thus, 7 y = 1680, so y = 240
Now x = = 420 and z = 540 – 2y = 60

9. Problem with straightforward approach
It is not efficient (e.g. computationally) when the number of equations is large

10. What can we do with systems of equations?
Example
Divide all by 10
Divide 2nd by 2

11. What can we do with systems of equations?
Example
Subtract 2nd from 3rd
Change the order of 1st and 2nd

12. What can we do with systems of equations?
Example
Multiply 1st by 7 and subtract from 2nd
Add 3rd to 2nd

13. Matrix form
Coefficient matrix
Augmented matrix

14. Matrix form
General 3x3 system

15. So, we can: Change the order of equations
Multiply an equation by a nonzero number
Add a multiple of one equation to another equation
Note: all this operations are invertible, i.e. the system can be returned to its original form by applying the same types of operations

16. Elementary row operations on augmented matrix
Switch two rows
Multiply a row by a nonzero number
Add a multiple of one row to another row
Note: all this operations are invertible, i.e. the matrix can be returned to its original form by applying the same types of operations

17. Notations Switch 1st and 2nd rows: R1  R2
Multiply 1st row by a nonzero number: aR1
Add a multiple of 1st row to the 2nd row
What are their inverses?

18. “Direct” and “Inverse” operations
Switch 1st and 2nd rows: R1  R2
Multiply 1st row by a nonzero number: aR1
Add a multiple of 1st row to the 2nd row: R2 → R2 + bR1
Inverse
Multiply 1st row by a nonzero number: (1/a)R1
Add a multiple of 1st row to the second row: R2 → R2 + (-b)R1

19. Goal of the operations
If possible, reduce the augmented matrix to the form:
Then x1=r1, x2=r2, x3=r3 is the (unique) solution

20. Linear systems and geometry
Systems of linear equations can be used to find intersections of planes and lines
Exercise Find the point at which the following planes intersect x+y+z=3, x+2y-z=3, 2x+3y+z=1